A083207 -id:A083207 - cp's OEIS frontend

A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090

Offset: 0

N. J. A. Sloane and J. H. Conway

See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.
a(n) is the least number N with n distinct prime factors (i.e., omega(N) = n, cf. A001221). - Lekraj Beedassy, Feb 15 2002
Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v, Jan 10 2004
Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - Lekraj Beedassy, Mar 31 2005
Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/Pi^2 < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie, Jun 11 2005
Let f be a multiplicative function with f(p) > f(p^k) > 1 (p prime, k > 1), f(p) > f(q) > 1 (p, q prime, p < q). Then the record maxima of f occur at n# for n >= 1. Similarly, if 0 < f(p) < f(p^k) < 1 (p prime, k > 1), 0 < f(p) < f(q) < 1 (p, q prime, p < q), then the record minima of f occur at n# for n >= 1. - David W. Wilson, Oct 23 2006
Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
Records in number of distinct prime divisors. - Artur Jasinski, Apr 06 2008
For n >= 2, the digital roots of a(n) are multiples of 3. - Parthasarathy Nambi, Aug 19 2009 [with corrections by Zak Seidov, Aug 30 2015]
Denominators of the sum of the ratios of consecutive primes (see A094661). - Vladimir Joseph Stephan Orlovsky, Oct 24 2009
Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i = 0..T(sqrt(N))} A005867(i)/A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010
The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. prime(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013
The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015
If a number k and a(n) are coprime and k < (prime(n+1))^b < a(n), where b is an integer, then k has fewer than b prime factors, counting multiplicity (i.e., bigomega(k) < b, cf. A001222). - Isaac Saffold, Dec 03 2017
If n > 0, then a(n) has 2^n unitary divisors (A034444), and a(n) is a record; i.e., if k < a(n) then k has fewer unitary divisors than a(n) has. - Clark Kimberling, Jun 26 2018
Unitary superabundant numbers: numbers k with a record value of the unitary abundancy index, A034448(k)/k > A034448(m)/m for all m < k. - Amiram Eldar, Apr 20 2019
Psi(n)/n is a new maximum for each primorial (psi = A001615) [proof in link: Patrick Sole and Michel Planat, proposition 1 page 2]; compare with comment 2004: Phi(n)/n is a new minimum for each primorial. - Bernard Schott, May 21 2020
The term "primorial" was coined by Harvey Dubner (1987). - Amiram Eldar, Apr 16 2021
a(n)^(1/n) is approximately (n log n)/e. - Charles R Greathouse IV, Jan 03 2023
Subsequence of A267124. - Frank M Jackson, Apr 14 2023

			a(9) = 23# = 2*3*5*7*11*13*17*19*23 = 223092870 divides the difference 5283234035979900 in the arithmetic progression of 26 primes A204189. - _Jonathan Sondow_, Jan 15 2012

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 49.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.
D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.

Alex Ermolaev, Table of n, a(n) for n = 0..350 (terms up to a(100) from T. D. Noe)
Iskander Aliev, Jesús De Loera, Fritz Eisenbrand, Timm Oertel, and Robert Weismantel, The Support of Integer Optimal Solutions, arXiv:1712.08923 [math.OC], 2017.
C. K. Caldwell, The Prime Glossary, Primorial.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
Harvey Dubner, Factorial and primorial primes, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197-203. (Annotated scanned copy)
F. Ellermann, Illustration for A002110, A005867, A038110, A060753.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, pp. 7, 81.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 7.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, preprint, CONCUR 2014 - Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp. 327-341.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, CONCUR 2014 - Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp. 327-341.
F. E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012.
Thomas Morrill, Further Development of "Non-Pythagorean" Musical Scales Based on Logarithms, arXiv:1804.08067 [math.HO], 2018.
J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.
Patrick Sole and Michel Planat, The Robin inequality for 7-free integers, INTEGERS, 2011, #A65.
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
G. Villemin's Almanach of Numbers, Primorielle.
Eric Weisstein's World of Mathematics, Primorial.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

A034386 gives the second version of the primorial numbers.
Subsequence of A005117 and of A064807. Apart from the first term, a subsequence of A083207.
Cf. A001615, A002182, A002201, A003418, A005235, A006862, A034444 (unitary divisors), A034448, A034387, A033188, A035345, A035346, A036691 (compositorial numbers), A049345 (primorial base representation), A057588, A060735 (and integer multiples), A061742 (squares), A072938, A079266, A087315, A094348, A106037, A121572, A053589, A064648, A132120, A260188.
Cf. A061720 (first differences), A143293 (partial sums).
Cf. also A276085, A276086.
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
-- Reinhard Zumkeller, Feb 19 2012, May 03 2011

[1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012

[1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015

A002110 := n -> mul(ithprime(i),i=1..n);

FoldList[Times, 1, Prime[Range[20]]]
primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial,20] (* José María Grau Ribas, Feb 15 2010 *)
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)

a(n)=prod(i=1,n, prime(i)) \\ Washington Bomfim, Sep 23 2008

p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) )  \\ Harry J. Smith, Nov 13 2009

a(n) = factorback(primes(n)) \\ David A. Corneth, May 06 2018

from sympy import primorial
def a(n): return 1 if n < 1 else primorial(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Mar 29 2017

[sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014

; with memoization-macro definec
(definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; Antti Karttunen, Aug 30 2016

Asymptotic expression for a(n): exp((1 + o(1)) * n * log(n)) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
a(n) = A054842(A002275(n)).
Binomial transform = A136104: (1, 3, 11, 55, 375, 3731, ...). Equals binomial transform of A121572: (1, 1, 3, 17, 119, 1509, ...). - Gary W. Adamson, Dec 14 2007
a(0) = 1, a(n+1) = prime(n)*a(n). - Juri-Stepan Gerasimov, Oct 15 2010
a(n) = Product_{i=1..n} A000040(i). - Jonathan Vos Post, Jul 17 2008
a(A051838(n)) = A116536(n) * A007504(A051838(n)). - Reinhard Zumkeller, Oct 03 2011
A000005(a(n)) = 2^n. - Carlos Eduardo Olivieri, Jun 16 2015
a(n) = A035345(n) - A005235(n) for n > 0. - Jonathan Sondow, Dec 02 2015
For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)). - Antti Karttunen, Aug 30 2016
A054841(a(n)) = A002275(n). - Michael De Vlieger, Aug 31 2016
a(n) = A270592(2*n+2) - A270592(2*n+1) if 0 <= n <= 4 (conjectured for all n by Alon Kellner). - Jonathan Sondow, Mar 25 2018
Sum_{n>=1} 1/a(n) = A064648. - Amiram Eldar, Oct 16 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = A132120. - Amiram Eldar, Apr 12 2021
Theta being Chebyshev's theta function, a(0) = exp(theta(1)), and for n > 0, a(n) = exp(theta(m)) for A000040(n) <= m < A000040(n+1) where m is an integer. - Miles Englezou, Nov 26 2024

A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

Original entry on oeis.org

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216

Offset: 1

N. J. A. Sloane

A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (this sequence), or deficient if sigma(k) < 2k (cf. A005100), where sigma(k) is the sum of the divisors of k (A000203).
The numbers 2^(p-1)*(2^p - 1) are perfect, where p is a prime such that 2^p - 1 is also prime (for the list of p's see A000043). There are no other even perfect numbers and it is believed that there are no odd perfect numbers.
Numbers k such that Sum_{d|k} 1/d = 2. - Benoit Cloitre, Apr 07 2002
For number of divisors of a(n) see A061645(n). Number of digits in a(n) is A061193(n). - Lekraj Beedassy, Jun 04 2004
All terms other than the first have digital root 1 (since 4^2 == 4 (mod 6), we have, by induction, 4^k == 4 (mod 6), or 2*2^(2*k) = 8 == 2 (mod 6), implying that Mersenne primes M = 2^p - 1, for odd p, are of the form 6*t+1). Thus perfect numbers N, being M-th triangular, have the form (6*t+1)*(3*t+1), whence the property N mod 9 = 1 for all N after the first. - Lekraj Beedassy, Aug 21 2004
The earliest recorded mention of this sequence is in Euclid's Elements, IX 36, about 300 BC. - Artur Jasinski, Jan 25 2006
Theorem (Euclid, Euler). An even number m is a perfect number if and only if m = 2^(k-1)*(2^k-1), where 2^k-1 is prime. Euler's idea came from Euclid's Proposition 36 of Book IX (see Weil). It follows that every even perfect number is also a triangular number. - Mohammad K. Azarian, Apr 16 2008
Triangular numbers (also generalized hexagonal numbers) A000217 whose indices are Mersenne primes A000668, assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008, Sep 15 2013
If a(n) is even, then 2*a(n) is in A181595. - Vladimir Shevelev, Nov 07 2010
Except for a(1) = 6, all even terms are of the form 30*k - 2 or 45*k + 1. - Arkadiusz Wesolowski, Mar 11 2012
a(4) = A229381(1) = 8128 is the "Simpsons's perfect number". - Jonathan Sondow, Jan 02 2015
Theorem (Farideh Firoozbakht): If m is an integer and both p and p^k-m-1 are prime numbers then x = p^(k-1)*(p^k-m-1) is a solution to the equation sigma(x) = (p*x+m)/(p-1). For example, if we take m=0 and p=2 we get Euclid's result about perfect numbers. - Farideh Firoozbakht, Mar 01 2015
The cototient of the even perfect numbers is a square; in particular, if 2^p - 1 is a Mersenne prime, cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2 (see A152921). So, this sequence is a subsequence of A063752. - Bernard Schott, Jan 11 2019
Euler's (1747) proof that all the even perfect number are of the form 2^(p-1)*(2^p-1) implies that their asymptotic density is 0. Kanold (1954) proved that the asymptotic density of odd perfect numbers is 0. - Amiram Eldar, Feb 13 2021
If k is perfect and semiprime, then k = 6. - Alexandra Hercilia Pereira Silva, Aug 30 2021
This sequence lists the fixed points of A001065. - Alois P. Heinz, Mar 10 2024

			6 is perfect because 6 = 1+2+3, the sum of all divisors of 6 less than 6; 28 is perfect because 28 = 1+2+4+7+14.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 2d ed. 1966, pp. 11-23.
Stanley J. Bezuszka, Perfect Numbers (Booklet 3, Motivated Math. Project Activities), Boston College Press, Chestnut Hill MA, 1980.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 136-137.
Euclid, Elements, Book IX, Section 36, about 300 BC.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.3 Perfect and Amicable Numbers, pp. 82-83.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 239.
T. Koshy, "The Ends Of A Mersenne Prime And An Even Perfect Number", Journal of Recreational Mathematics, Baywood, NY, 1998, pp. 196-202.
Joseph S. Madachy, Madachy's Mathematical Recreations, New York: Dover Publications, Inc., 1979, p. 149 (First publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation).
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 46-48, 244-245.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 83-87.
József Sándor and Borislav Crstici, Handbook of Number Theory, II, Springer Verlag, 2004.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ian Stewart, L'univers des nombres, "Diviser Pour Régner", Chapter 14, pp. 74-81, Belin-Pour La Science, Paris, 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, chapter 4, pages 127-149.
Horace S. Uhler, On the 16th and 17th perfect numbers, Scripta Math., Vol. 19 (1953), pp. 128-131.
André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, p. 6.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 107-110, Penguin Books, 1987.

E-Hern Lee, Table of n, a(n) for n = 1..15 (terms 1-14 from David Wasserman)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Perfect Numbers. [broken link]
Anonymous, Timetable of discovery of perfect numbers. [broken link]
Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Richard P. Brent and Graeme L. Cohen, A new lower bound for odd perfect numbers, Math. Comp., Vol. 53, No. 187 (1989), pp. 431-437, S7; alternative link.
Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, A new approach to lower bounds for odd perfect numbers, Report TR-CS-88-08, CSL, ANU, August 1988, 71 pp.
Richard P. Brent, Graeme L. Cohen and Herman J. J. te Riele, Improved Techniques For Lower Bounds For Odd Perfect Numbers, Math. Comp., Vol. 57, No. 196 (1991), pp. 857-868.
J. Britton, Perfect Number Analyser.
Chris K. Caldwell, Perfect number.
Chris K. Caldwell, Mersenne Primes, etc.
Chris K. Caldwell, Iterated sums of the digits of a perfect number converge to 1.
Jose Arnaldo B. Dris, The Abundancy Index of Divisors of Odd Perfect Numbers, J. Int. Seq., Vol. 15 (2012) Article # 12.4.4.
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), p. 243.
Roger B. Eggleston, Equisum Partitions of Sets of Positive Integers, Algorithms, Vol. 12, No. 8 (2019), Article 164.
Leonhard Euler, De numeris amicibilibus>, Commentationes arithmeticae collectae, Vol. 2 (1849), pp. 627-636. Written in 1747.
Bakir Farhi, On the representation of an even perfect number as the sum of a limited number of cubes, arXiv:1504.07322 [math.NT], 2015.
Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.1.
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 111.
J. W. Gaberdiel, A Study of Perfect Numbers and Related Topics.
Takeshi Goto and Yasuo Ohno, Largest prime factor of an odd perfect number, 2006.
Kevin G. Hare, New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp., Vol. 76, No. 260 (2007), pp. 2241-2248; arXiv preprint, arXiv:math/0501070 [math.NT], 2005-2006.
Azizul Hoque and Himashree Kalita, Generalized perfect numbers connected with arithmetic functions, Math. Sci. Lett., Vol. 3, No. 3 (2014), pp. 249-253.
C.-E. Jean, "Recreomath" Online Dictionary, Nombre parfait.
Hans-Joachim Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, The Ramanujan Journal, Vol. 46, No. 3 (2018), pp. 633-655; arXiv preprint, arXiv:1610.07793 [math.NT], 2016.
Pedro Laborde, A Note on the Even Perfect Numbers, The American Mathematical Monthly, Vol. 62, No. 5 (May, 1955), pp. 348-349 (2 pages).
Tom Leinster, Perfect numbers and groups, arXiv:math/0104012 [math.GR], 2001.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
Daniel Lustig, The algebraic independence of the sum of divisors functions, Journal of Number Theory, Volume 130, Issue 11 (November 2010), pp. 2628-2633.
T. Masiwa, T. Shonhiwa & G. Hitchcock, Perfect Numbers & Mersenne Primes.
Mathforum, Perfect Numbers.
Mathforum, List of Perfect Numbers.
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seqs., Vol. 3 (2000), Article #00.1.3.
Gérard P. Michon, Perfect Numbers, Mersenne Primes.
David Moews, Perfect, amicable and sociable numbers.
Derek Muller, The Oldest Unsolved Problem in Math, Veritasium, YouTube video, 2024.
Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Mathematics of Computation, Vol. 76, No. 260 (2007), pp. 2109-2126; arXiv preprint, arXiv:math/0602485 [math.NT], 2006.
Walter Nissen, Abundancy : Some Resources , 2008-2010.
J. J. O'Connor and E. F. Robertson, Perfect Numbers.
J. O. M. Pedersen, Perfect numbers. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Ivars Peterson, Cubes of Perfection, MathTrek, 1998.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
D. Romagnoli, Perfect Numbers (Text in Italian). [From _Lekraj Beedassy_, Jun 26 2009]
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
D. Scheffler and R. Ondrejka, The numerical evaluation of the eighteenth perfect number, Math. Comp., Vol. 14, No. 70 (1960), pp. 199-200.
K. Schneider, perfect number, PlanetMath.org.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240; arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
G. Villemin's Almanach of Numbers, Nombres Parfaits.
J. Voight, Perfect Numbers:An Elementary Introduction.
Eric Weisstein's World of Mathematics, Perfect Number.
Eric Weisstein's World of Mathematics, Odd Perfect Number.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Perfect number.
Tomohiro Yamada, On the divisibility of odd perfect numbers by a high power of a prime, arXiv:math/0511410 [math.NT], 2005-2007.
Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 1.
Index entries for "core" sequences
Index entries for sequences where any odd perfect numbers must occur

See A000043 for the current state of knowledge about Mersenne primes.
Cf. A007539, A005820, A027687, A046060, A046061, A000668, A090748, A133033, A000217, A000384, A019279, A061652, A006516, A144912, A153800, A007593, A220290, A028499-A028502, A034916, A065549, A275496, A063752, A156552, A152921, A324201.
Cf. A228058 for Euler's criterion for odd terms.
Positions of 0's in A033879 and in A033880.
Subsequence of following sequences: A005835, A006039, A007691, A023196, A043305, A065997, A083207, A109510, A118372, A216782, A246282, A263837, A294900, A333646, A334410, A335267, A336702, A341622, A342922, A344755, A352739, A357462, and (the even terms), of: A005153, A063752, A174973, A336547, A338520.
Cf. A001065.

a000396 n = a000396_list !! (n-1)
a000396_list = [x | x <- [1..], a000203 x == 2 * x]
-- Reinhard Zumkeller, Jan 20 2012

Select[Range[9000], DivisorSigma[1,#]== 2*# &] (* G. C. Greubel, Oct 03 2017 *)
PerfectNumber[Range[15]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 10 2018 *)

PARI
```
isA000396(n) = (sigma(n) == 2*n);
```

from sympy import divisor_sigma
def ok(n): return n > 0 and divisor_sigma(n) == 2*n
print([k for k in range(9999) if ok(k)]) # Michael S. Branicky, Mar 12 2022

The perfect number N = 2^(p-1)*(2^p - 1) is also multiplicatively p-perfect (i.e., A007955(N) = N^p), since tau(N) = 2*p. - Lekraj Beedassy, Sep 21 2004
a(n) = 2^A133033(n) - 2^A090748(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 28 2008
a(n) = A000668(n)*(A000668(n)+1)/2, assuming there are no odd perfect numbers. - Omar E. Pol, Apr 23 2008
a(n) = A000217(A000668(n)), assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008
a(n) = Sum of the first A000668(n) positive integers, assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008
a(n) = A000384(A019279(n)), assuming there are no odd perfect numbers and no odd superperfect numbers. a(n) = A000384(A061652(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Aug 17 2008
a(n) = A006516(A000043(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Aug 30 2008
From Reikku Kulon, Oct 14 2008: (Start)
A144912(2, a(n)) = 1;
A144912(4, a(n)) = -1 for n > 1;
A144912(8, a(n)) = 5 or -5 for all n except 2;
A144912(16, a(n)) = -4 or -13 for n > 1. (End)
a(n) = A019279(n)*A000668(n), assuming there are no odd perfect numbers and odd superperfect numbers. a(n) = A061652(n)*A000668(n), assuming there are no odd perfect numbers. - Omar E. Pol, Jan 09 2009
a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Jan 14 2009
Even perfect numbers N = K*A000203(K), where K = A019279(n) = 2^(p-1), A000203(A019279(n)) = A000668(n) = 2^p - 1 = M(p), p = A000043(n). - Lekraj Beedassy, May 02 2009
a(n) = A060286(A016027(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 13 2012
For n >= 2, a(n) = Sum_{k=1..A065549(n)} (2*k-1)^3, assuming there are no odd perfect numbers. - Derek Orr, Sep 28 2013
a(n) = A275496(2^((A000043(n) - 1)/2)) - 2^A000043(n), assuming there are no odd perfect numbers. - Daniel Poveda Parrilla, Aug 16 2016
a(n) = A156552(A324201(n)), assuming there are no odd perfect numbers. - Antti Karttunen, Mar 28 2019
a(n) = ((2^(A000043(n)))^3 - (2^(A000043(n)) - 1)^3 - 1)/6, assuming there are no odd perfect numbers. - Jules Beauchamp, Jun 06 2025

I removed a large number of comments that assumed there are no odd perfect numbers. There were so many it was getting hard to tell which comments were true and which were conjectures. - N. J. A. Sloane, Apr 16 2023

Reference to Albert H. Beiler's book updated by Harvey P. Dale, Jan 13 2025

A002182 Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160

Offset: 1

N. J. A. Sloane

Where record values of d(n) occur: d(n) > d(k) for all k < n.
A002183 is the RECORDS transform of A000005, i.e., lists the corresponding values d(n) for n in A002182.
Flammenkamp's page also has a copy of the missing Siano paper.
Highly composite numbers are the product of primorials, A002110. See A112779 for the number of primorial terms in the product of a highly composite number. - Jud McCranie, Jun 12 2005
Sigma and tau for highly composite numbers through the 146th entry conform to a power fit as follows: log(sigma)=A*log(tau)^B where (A,B) =~ (1.45,1.38). - Bill McEachen, May 24 2006
a(n) often corresponds to P(n,m) = number of permutations of n things taken m at a time. Specifically, if start=1, pointers 1-6, 9, 10, 13-15, 17-19, 22, 23, 28, 34, 37, 43, 52, ... An example is a(37)=665280, which is P(12,6)=12!/(12-6)!. - Bill McEachen, Feb 09 2009
Concerning the previous comment, if m=1, then P(n,m) can represent any number. So let's assume m > 1. Searching the first 1000 terms, the only indices of terms of the form P(n,m) are 4, 5, 6, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 27, 28, 31, 34, 37, 41, 43, 44, 47, 50, 52, and 54. Note that a(44) = 4324320 = P(2079,2). See A163264. - T. D. Noe, Jun 10 2009
A large number of highly composite numbers have 9 as their digit root. - Parthasarathy Nambi, Jun 07 2009
Because 9 divides all highly composite numbers greater than 1680, those numbers have digital root 9. - T. D. Noe, Jul 24 2009
See A181309 for highly composite numbers that are not highly abundant.
a(n) is also defined by the recurrence: a(1) = 1, a(n+1)/sigma(a(n+1)) < a(n) / sigma(a(n)). - Michel Lagneau, Jan 02 2012 [NOTE: This "definition" is wrong (a(20)=7560 does not satisfy this inequality) and incomplete: It does not determine a sequence uniquely, e.g., any subsequence would satisfy the same relation. The intended meaning is probably the definition of the (different) sequence A004394. - M. F. Hasler, Sep 13 2012]
Up to a(1000), the terms beyond a(5) = 12 resp. beyond a(9) = 60 are a multiples of these. Is this true for all subsequent terms? - M. F. Hasler, Sep 13 2012 [Yes: see EXAMPLE in A199337! - M. F. Hasler, Jan 03 2020]
Differs from the superabundant numbers from a(20)=7560 on, which is not in A004394. The latter is not a subsequence of A002182, as might appear from considering the displayed terms: The two sequences have only 449 terms in common, the largest of which is A002182(2567) = A004394(1023). See A166735 for superabundant numbers that are not highly composite, and A004394 for further information. - M. F. Hasler, Sep 13 2012
Subset of A067128 and of A025487. - David A. Corneth, May 16 2016, Jan 03 2020
It seems that a(n) +- 1 is often prime. For n <= 1000 there are 210 individual primes and 17 pairs of twin primes. See link to Lim's paper below. - Dmitry Kamenetsky, Mar 02 2019
There are infinitely many numbers in this sequence and a(n+1) <= 2*a(n), because it is sufficient to multiply a(n) by 2 to get a number having more divisors. (This proves Guess 0 in the Lim paper.) For n = (1, 2, 4, 5, 9, 13, 18, ...) one has equality in this bound, but asymptotically a(n+1)/a(n) goes to 1, cf. formula due to Erdős. See A068507 for the terms such that a(n)+-1 are twin primes. - M. F. Hasler, Jun 23 2019
Conjecture: For n > 7, a(n) is a Zumkeller number (A083207). Verified for n up to and including 48. If this conjecture is true, one may base on it an alternative proof of the fact that for n>7 a(n) is not a perfect square (see Fact 5, Rao/Peng arXiv link at A083207). - Ivan N. Ianakiev, Jun 29 2019
The conjecture above is true (see the proof in the "Links" section). - Ivan N. Ianakiev, Jan 31 2020
The first instance of omega(a(n)) < omega(a(n-1)) (omega = A001221: number of prime divisors) is at a(26) = 45360. Up to n = 10^4, 1759 terms have this property, but omega decreases by 2 only at indices n = 5857, 5914 and 5971. - M. F. Hasler, Jan 02 2020
Inequality (54) in Ramanujan (1915) implies that for any m there is n* such that m | a(n) for all n > n*: see A199337 for the proof. - M. F. Hasler, Jan 03 2020

			a(5) = 12 is in the sequence because A000005(12) is larger than any earlier value in A000005. - _M. F. Hasler_, Jan 03 2020

CRC Press Standard Mathematical Tables, 28th Ed, p. 61.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris 2008.
L. E. Dickson, History of Theory of Numbers, I, p. 323.
Ross Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985
Jean-Louis Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 88.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (obtained from A. Flammenkamp's data; first 1000 terms from T. D. Noe)
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Harold W. Ellingsen, Jr., A Fresh Look at Highly Composite Numbers, The American Mathematical Monthly, Vol. 126, No. 8 (2019), pp. 740-741.
Paul Erdős, On Highly composite numbers, J. London Math. Soc., Vol. 19 (1944), pp. 130-133, MR7,145d; Zentralblatt 61,79.
Achim Flammenkamp, Highly composite numbers.
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Achim Flammenkamp, List of the first 779,674 highly composite numbers.
James Grime and Brady Haran, 5040 and other Anti-Prime Numbers, Numberphile video (2016).
Bob Hinman, Letter to N. J. A. Sloane, Aug. 1980.
Ivan N. Ianakiev, On the question "Which Highly composite numbers (A002182) are Zumkeller numbers (A083207)?".
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
Aneesh M. Koya and P. P. Deepthi, Plug and play self-configurable IoT gateway node for telemonitoring of ECG, Computers in Biology and Medicine, Vol. 112 (2019), 103359.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543; arXiv:math/0008177 [math.NT], 2000-2001.
Bill Lauritzen, Versatile Numbers -Versatile Economics.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola Magazine, Volume 54, Issue 3, (2018).
R. J. Mathar, Maple program to convert the Flammenkamp file to an OEIS b-file.
R. J. Mathar, Output of above Maple program. [Uncompresses to 9.1 MB]
Graeme McRae, Highly Composite Numbers.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers' (Text in French).
Jean-Louis Nicolas and Guy Robin, Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
Kevin O'Bryant, PlanetMath.org, Highly composite number.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915), pp. 347-409. (DOI: 10.1112/plms/s2_14.1.347, also available with an additional footnote in the PDF at http://ramanujan.sirinudi.org/Volumes/published/ram15.html)
Steven Ratering, An interesting subset of the highly composite numbers, Math. Mag., Vol. 64, No. 5 (1991), pp. 343-346.
Guy Robin, Méthodes d'optimisation pour un problème de théorie des nombres, RAIRO Informatique Théorique, Vol. 17, No. 3 (1983), pp. 239-247.
Vladimir Shevelev, On Erdős constant, arXiv:1605.08884 [math.NT], 2016.
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.
N. J. A. Sloane, Transforms.
Michel Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Highly Composite Number.
Wikipedia, Highly composite number.

Cf. A000005 (number of divisors), A002110, A002183, A002473, A004394, A025487, A106037, A108602, A112778, A112779, A112780, A112781, A006218, A126098, A002201, A072938, A094348, A003418, A161184, A037992 (infinitary analog), A108951, A329902, A352418.
Cf. A261100 (a left inverse).
Cf. A002808. - Peter J. Marko, Aug 16 2018
Cf. A279930 (highly composite and highly Brazilian).
Cf. A068507 (terms such that a(n)+-1 are twin primes).
Cf. A199337 (number of terms not divisible by n).

a = 0; Do[b = DivisorSigma[0, n]; If[b > a, a = b; Print[n]], {n, 1, 10^7}]
(* Convert A. Flammenkamp's 779674-term dataset; first, decompress, rename "HCN.txt": *)
a = Times @@ {Times @@ Prime@ Range@ ToExpression@ First@ #1, If[# == {}, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, #]] &@ DeleteCases[-1 + Flatten@ Map[If[StringFreeQ[#, "^"], ToExpression@ #, ConstantArray[#1, #2] & @@ ToExpression@ StringSplit[#, "^"]] &, #2], 0]} & @@ TakeDrop[StringSplit@ #, 1] & /@ Import["HCN.txt", "Data"] (* Michael De Vlieger, May 08 2018 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,2163000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[All,1]] (* Harvey P. Dale, May 13 2022 *)
NestList[Function[last,
  Module[{d = DivisorSigma[0, last]},
   NestWhile[# + 1 &, last, DivisorSigma[0, #] <= d &]]], 1, 40] (* Steven Lu, Mar 30 2023 *)

print1(r=1); forstep(n=2,1e5,2, if(numdiv(n)>r, r=numdiv(n); print1(", "n))) \\ Charles R Greathouse IV, Jun 10 2011

v002182 = [1]/*vector for memoization*/; A002182(n, i = #v002182) ={ if(n > i, v002182 = Vec(v002182, n); my(k = v002182[i], d, s=1); until(i == n, d = numdiv(k); s<60 && k>=60 && s=60; until(numdiv(k += s) > d,); v002182[i++] = k); k, v002182[n])} \\ Antti Karttunen, Jun 06 2017; edited by M. F. Hasler, Jan 03 2020 and Jun 20 2022

is_A002182(n, a=1, b=1)={while(n>A002182(b*=2), a*=2); until(a>b, my(m=(a+b)\2, t=A002182(m)); if(tn, b=m-1, return(m)))} \\ Also used in other sequences. - M. F. Hasler, Jun 20 2022

from sympy import divisor_count
A002182_list, r = [], 0
for i in range(1,10**4):
    d = divisor_count(i)
    if d > r:
        r = d
        A002182_list.append(i) # Chai Wah Wu, Mar 23 2015

Also, for n >= 2, smallest values of p for which A006218(p)-A006318(p-1) = A002183(n). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n+1) < a(n) * (1+log(a(n))^-c) for some positive c (see Erdős). - David A. Corneth, May 16 2016
a(n) = A108951(A329902(n)). - Antti Karttunen, Jan 08 2020
a(n+1) <= 2*a(n). For cases where the equal sign holds, see A072938. - A.H.M. Smeets, Jul 10 2021
Sum_{n>=1} 1/a(n) = A352418. - Amiram Eldar, Mar 24 2022

Jun 19 1996: Changed beginning to start at 1.
Jul 10 1996: Matthew Conroy points out that these are different from the super-abundant numbers - see A004394. Last 8 terms sent by J. Lowell; checked by Jud McCranie.
Description corrected by Gerard Schildberger and N. J. A. Sloane, Apr 04 2001
Additional references from Lekraj Beedassy, Jul 24 2001

A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 0

Roland Anderson (roland.anderson(AT)swipnet.se)

The minimal exponent of the symmetric group S_n, i.e., the least positive integer for which x^a(n)=1 for all x in S_n. - Franz Vrabec, Dec 28 2008
Product over all primes of highest power of prime less than or equal to n. a(0) = 1 by convention.
Also smallest number whose set of divisors contains an n-term arithmetic progression. - Reinhard Zumkeller, Dec 09 2002
An assertion equivalent to the Riemann hypothesis is: | log(a(n)) - n | < sqrt(n) * log(n)^2. - Lekraj Beedassy, Aug 27 2006. (This is wrong for n = 1 and n = 2. Should "for n large enough" be added? - Georgi Guninski, Oct 22 2011)
Corollary 3 of Farhi gives a proof that a(n) >= 2^(n-1). - Jonathan Vos Post, Jun 15 2009
Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. - Mats Granvik, Jul 08 2009
Greg Martin (see link) proved that "the product of the Gamma function sampled over the set of all rational numbers in the open interval (0,1) whose denominator in lowest terms is at most n" equals (2*Pi)^(1/2)*a(n)^(-1/2). - Jonathan Vos Post, Jul 28 2009
a(n) = lcm(A188666(n), A188666(n)+1, ..., n). - Reinhard Zumkeller, Apr 25 2011
a(n+1) is the smallest integer such that all polynomials a(n+1)*(1^i + 2^i + ... + m^i) in m, for i=0,1,...,n, are polynomials with integer coefficients. - Vladimir Shevelev, Dec 23 2011
It appears that A020500(n) = a(n)/a(n-1). - Asher Auel, corrected by Bill McEachen, Apr 05 2024
n-th distinct value = A051451(n). - Matthew Vandermast, Nov 27 2009
a(n+1) = least common multiple of n-th row in A213999. - Reinhard Zumkeller, Jul 03 2012
For n > 2, (n-1) = Sum_{k=2..n} exp(a(n)*2*i*Pi/k). - Eric Desbiaux, Sep 13 2012
First column minus second column of A027446. - Eric Desbiaux, Mar 29 2013
For n > 0, a(n) is the smallest number k such that n is the n-th divisor of k. - Michel Lagneau, Apr 24 2014
Slowest growing integer > 0 in Z converging to 0 in Z^ when considered as profinite integer. - Herbert Eberle, May 01 2016
What is the largest number of consecutive terms that are all equal? I found 112 equal terms from a(370261) to a(370372). - Dmitry Kamenetsky, May 05 2019
Answer: there exist arbitrarily long sequences of consecutive terms with the same value; also, the maximal run of consecutive terms with different values is 5 from a(1) to a(5) (see link Roger B. Eggleton). - Bernard Schott, Aug 07 2019
Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: a(A329570(m))^2 is a (not minimal) bound above which all highly composite numbers are divisible by m, according to the right part of that inequality. - M. F. Hasler, Jan 04 2020
For n > 2, a(n) is of the form 2^e_1 * p_2^e_2 * ... * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) <= e_1. Therefore, 2^e * p_m^e_m is a primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 2, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m, is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 10 2020
For n > 1, 2|(a(n)+2) ... n|(a(n)+n), so a(n)+2 .. a(n)+n are all composite and (part of) a prime gap of at least n.  (Compare n!+2 .. n!+n). - Stephen E. Witham, Oct 09 2021

			LCM of {1,2,3,4,5,6} = 60. The primes up to 6 are 2, 3 and 5. floor(log(6)/log(2)) = 2 so the exponent of 2 is 2.
floor(log(6)/log(3)) = 1 so the exponent of 3 is 1.
floor(log(6)/log(5)) = 1 so the exponent of 5 is 1. Therefore, a(6) = 2^2 * 3^1 * 5^1 = 60. - _David A. Corneth_, Jun 02 2017

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..2308 (first 501 terms from T. D. Noe)
R. Anderson and N. J. A. Sloane, Correspondence, 1975.
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Javier Cilleruelo, Juanjo Rué, Paulius Šarka, and Ana Zumalacárregui, The least common multiple of sets of positive integers, arXiv:1112.3013 [math.NT], 2011.
R. E. Crandall and C. Pomerance, Prime numbers: a computational perspective, MR2156291, p. 61.
Roger B. Eggleton, Least Common Multiple of {1,2,...,n}, Mathematics Magazine, 61(1) (1988), pp. 47-48, Problem 1252.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295 [math.NT], 2009.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly 116(9) (2009), 836-839.
Steven Finch, Cilleruelo's LCM Constants, 2013. [Cached copy, with permission of the author]
V. L. Gavrikov, On property of least common multiple to be a D-magic number, arXiv:1806.09264 [math.NT], 2018.
S. Labbé and E. Pelantová, Palindromic sequences generated from marked morphisms, arXiv:1409.7510 [math.CO], 2014.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534-543. arXiv:math/0008177 [math.NT], 2000-2001.
Peter Luschny and S. Wehmeier, The lcm(1, 2, ..., n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99, pp 213-219 (2015).
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
M. Nair, On Chebychev-type inequalities for primes Amer. Math. Monthly 89(2) (1982), 126-129.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society ser. 2, vol. XIV, no. 1 (1915), pp 347-409. (A variant of a better quality with an additional footnote is available here.)
E. S. Selmer, On the number of prime divisors of a binomial coefficient, Math. Scand. 39 (1976), no. 2, 271-281 (1977).
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. AMS 131 (2003), 3335.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
M. Tchebichef, Mémoire sur les nombres premiers, J. Math. Pures Appliquées 17 (1852), 366-390.
Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1) (1901), 159-182.
Eric Weisstein's World of Mathematics, Least Common Multiple, Chebyshev Functions, Mangoldt Function.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to lcm's

Row products of A133233.
Cf. A000142, A000793, A002110, A002182, A002201, A002944, A014963, A020500, A025527, A038610, A051173, A064446, A064859, A069513, A072938, A093880, A094348, A096179, A099996, A102910, A106037, A119682, A179661, A193181, A225558, A225630, A225632, A225640, A225642.
Cf. A025528 (number of prime factors of a(n) with multiplicity).
Cf. A275120 (lengths of runs of consecutive equal terms), A276781 (ordinal transform from term a(1)=1 onward).

a003418 = foldl lcm 1 . enumFromTo 2
-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011

[1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013

[Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015

A003418 := n-> lcm(seq(i,i=1..n));
HalfFarey := proc(n) local a,b,c,d,k,s; a := 0; b := 1; c := 1; d := n; s := NULL; do k := iquo(n + b, d); a, b, c, d := c, d, k*c - a, k*d - b; if 2*a > b then break fi; s := s,(a/b); od: [s] end: LCM := proc(n) local i; (1/2)*mul(2*sin(Pi*i),i=HalfFarey(n))^2 end: # Peter Luschny
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jun 10 2021

Table[LCM @@ Range[n], {n, 1, 40}] (* Stefan Steinerberger, Apr 01 2006 *)
FoldList[ LCM, 1, Range@ 28]
A003418[0] := 1; A003418[1] := 1; A003418[n_] := A003418[n] = LCM[n,A003418[n-1]]; (* Enrique Pérez Herrero, Jan 08 2011 *)
Table[Product[Prime[i]^Floor[Log[Prime[i], n]], {i, PrimePi[n]}], {n, 0, 28}] (* Wei Zhou, Jun 25 2011 *)
Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}] (* Fred Daniel Kline, May 22 2014 *)
a1[n_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1,1+n/2] - PolyGamma[1,(1+n)/2])) // Simplify
a[n_] := Denominator[Sqrt[a1[n]]];
Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[[2]]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *)

a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t

a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))

a(n)=my(v=primes(primepi(n)),k=sqrtint(n),L=log(n+.5));prod(i=1,#v,if(v[i]>k,v[i],v[i]^(L\log(v[i])))) \\ Charles R Greathouse IV, Dec 21 2011

a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via Charles R Greathouse IV]

n=1; lim=100; i=1; j=1; until(n==lim, a=lcm(j,i+1); i++; j=a; n++; print(n" "a);); \\ Mike Winkler, Sep 07 2013

from functools import reduce
from operator import mul
from sympy import sieve
def integerlog(n,b): # find largest integer k>=0 such that b^k <= n
    kmin, kmax = 0,1
    while b**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if b**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmin
def A003418(n):
    return reduce(mul,(p**integerlog(n,p) for p in sieve.primerange(1,n+1)),1) # Chai Wah Wu, Mar 13 2021

# generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return [1] + list(accumulate(range(1, nn+1), lcm))
print(aupton(30)) # Michael S. Branicky, Jun 10 2021

[lcm(range(1,n)) for n in range(1, 30)] # Zerinvary Lajos, Jun 06 2009

(define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018

The prime number theorem implies that lcm(1,2,...,n) = exp(n(1+o(1))) as n -> infinity. In other words, log(lcm(1,2,...,n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005
a(n) = Product (p^(floor(log n/log p))), where p runs through primes not exceeding n (i.e., primes 2 through A007917(n)). - Lekraj Beedassy, Jul 27 2004
Greg Martin showed that a(n) = lcm(1,2,3,...,n) = Product_{i = Farey(n), 0 < i < 1} 2*Pi/Gamma(i)^2. This can be rewritten (for n > 1) as a(n) = (1/2)*(Product_{i = Farey(n), 0 < i <= 1/2} 2*sin(i*Pi))^2. - Peter Luschny, Aug 08 2009
Recursive formula useful for computations: a(0)=1; a(1)=1; a(n)=lcm(n,a(n-1)). - Enrique Pérez Herrero, Jan 08 2011
From Enrique Pérez Herrero, Jun 01 2011: (Start)
a(n)/a(n-1) = A014963(n).
if n is a prime power p^k then a(n)=a(p^k)=p*a(n-1), otherwise a(n)=a(n-1).
a(n) = Product_{k=2..n} (1 + (A007947(k)-1)*floor(1/A001221(k))), for n > 1. (End)
a(n) = A079542(n+1, 2) for n > 1.
a(n) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)). - Peter Luschny, Sep 01 2012
a(n) = A025529(n) - A027457(n). - Eric Desbiaux, Mar 14 2013
a(n) = exp(Psi(n)) = 2 * Product_{k=2..A002088(n)} (1 - exp(2*Pi*i * A038566(k+1) / A038567(k))), where i is the imaginary unit, and Psi the second Chebyshev's function. - Eric Desbiaux, Aug 13 2014
a(n) = A064446(n)*A038610(n). - Anthony Browne, Jun 16 2016
a(n) = A000142(n) / A025527(n) = A000793(n) * A225558(n). - Antti Karttunen, Jun 02 2017
log(a(n)) = Sum_{k>=1} (A309229(n, k)/k - 1/k). - Mats Granvik, Aug 10 2019
From Petros Hadjicostas, Jul 24 2020: (Start)
Nair (1982) proved that 2^n <= a(n) <= 4^n for n >= 9. See also Farhi (2009). Nair also proved that
a(n) = lcm(m*binomial(n,m): 1 <= m <= n) and
a(n) = gcd(a(m)*binomial(n,m): n/2 <= m <= n). (End)
Sum_{n>=1} 1/a(n) = A064859. - Bernard Schott, Aug 24 2020

A003586 3-smooth numbers: numbers of the form 2^i*3^j with i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3456, 3888

Offset: 1

Paul Zimmermann, Dec 11 1996

This sequence is easily confused with A033845, which gives numbers of the form 2^i*3^j with i, j >= 1. Don't simply say "numbers of the form 2^i*3^j", but specify which sequence you mean. - N. J. A. Sloane, May 26 2024
These numbers were once called "harmonic numbers", see Lenstra links. - N. J. A. Sloane, Jul 03 2015
Successive numbers k such that phi(6k) = 2k. - Artur Jasinski, Nov 05 2008
Where record values greater than 1 occur in A088468: A160519(n) = A088468(a(n)). - Reinhard Zumkeller, May 16 2009
Also numbers that are divisible by neither 6k - 1 nor 6k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010
Also numbers m such that the rooted tree with Matula-Goebel number m has m antichains. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. An antichain is a nonempty set of mutually incomparable vertices. Example: m=4 is in the sequence because the corresponding rooted tree is \/=ARB (R is the root) having 4 antichains (A, R, B, AB). - Emeric Deutsch, Jan 30 2012
A204455(3*a(n)) = 3, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
The number of terms less than or equal to n is Sum_{i=0..floor(log_2(n))} floor(log_3(n/2^i) + 1), or Sum_{i=0..floor(log_3(n))} floor(log_2(n/3^i) + 1), which requires fewer terms to compute. - Robert G. Wilson v, Aug 17 2012
Named 3-friables in French. - Michel Marcus, Jul 17 2013
In the 14th century Levi Ben Gerson proved that the only pairs of terms which differ by 1 are (1,2), (2,3), (3,4), and (8,9); see A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014
Range of values of A000005(n) (and also A181819(n)) for cubefree numbers n. - Matthew Vandermast, May 14 2014
A036561 is a permutation of this sequence. - L. Edson Jeffery, Sep 22 2014
Also the sorted union of A000244 and A007694. - Lei Zhou, Apr 19 2017
The sum of the reciprocals of the 3-smooth numbers is equal to 3. Brief proof: 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + ... = (Sum_{k>=0} 1/2^k) * (Sum_{m>=0} 1/3^m) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3. - Bernard Schott, Feb 19 2019
Also those integers k for which, for every prime p > 3, p^(2k) - 1 == 0 (mod 24k). - Federico Provvedi, May 23 2022
For n>1, the exponents’ parity {parity(i), parity(j)} of one out of four consecutive terms is {odd, odd}. Therefore, for n>1, at least one out of every four consecutive terms is a Zumkeller number (A083207). If for the term whose parity is {even, odd}, even also means nonzero, then this term is also a Zumkeller number (as is the case with the last of the four consecutive terms 1296, 1458, 1536, 1728). - Ivan N. Ianakiev, Jul 10 2022
Except the initial terms 2, 3, 4, 8, 9 and 16, these are numbers k such that k^6 divides 6^k. Except the initial terms 2, 3, 4, 6, 8, 9, 16, 18 and 27, these are numbers k such that k^12 divides 12^k. - Mohammed Yaseen, Jul 21 2022
In music theory, a comma is a ratio, close to 1 (typically less than 1.04), between two natural numbers divisible by only small primes (typically single digit). In this sequence, a(131) / a(130) = 531441 / 524288 ~ 1.013643 is the Pythagorean comma (A221363), the difference between 12 perfect fifths and 7 octaves. - Hal M. Switkay, Mar 23 2025

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 654 pp. 85, 287-8, Ellipses Paris 2004.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.
R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

Lei Zhou, Table of n, a(n) for n = 1..10000 (first 501 terms from Franklin T. Adams-Watters)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
Benoit Cloitre, a(n)/((1/sqrt(6))*exp(sqrt(2*log(2)*log(3)*n))) for 0

Natalia da Silva, Serban Raianu, and Hector Salgado, Differences of Harmonic Numbers and the abc-Conjecture, arXiv:1708.00620 [math.NT], 2017.
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 252. Book's website
H. W. Lenstra Jr., Harmonic Numbers
H. W. Lenstra, Jr., Harmonic Numbers and the ABC-conjecture, Abstract of talk, May 30, 2001 [Annotated scanned copy]
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
D. J. Mintz, 2,3 sequence as a binary mixture, Fib. Quarterly, Vol. 19, No 4, Oct 1981, pp. 351-360.
I. Peterson, Medieval Harmony
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A051037, A002473, A051038, A080197, A080681, A080682, A117221, A105420, A062051, A117222, A117220, A090184, A131096, A131097, A186711, A186712, A186771, A088468, A061987, A080683 (p-smooth numbers with other values of p), A025613 (a subsequence).
Cf. also A235365, A235366, A236210, A036561.
Cf. also A000244, A007694. - Lei Zhou, Apr 19 2017
Cf. A191475 (successive values of i), A191476 (successive values of j), A022330 (indices of the pure terms 2^i), A022331 (indices of the pure terms 3^j). - N. J. A. Sloane, May 26 2024
Cf. A221363.

import Data.Set (Set, singleton, insert, deleteFindMin)
smooth :: Set Integer -> [Integer]
smooth s = x : smooth (insert (3*x) $ insert (2*x) s')
  where (x, s') = deleteFindMin s
a003586_list = smooth (singleton 1)
a003586 n = a003586_list !! (n-1)
-- Reinhard Zumkeller, Dec 16 2010

[n: n in [1..4000] | PrimeDivisors(n) subset [2,3]]; // Bruno Berselli, Sep 24 2012

A003586 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do numtheory[factorset](a) minus {2,3} ; if % = {} then return a; end if; end do: end if; end proc: # R. J. Mathar, Feb 28 2011
with(numtheory): for i from 1 to 23328 do if(i/phi(i)=3)then print(i/6) fi od; # Gary Detlefs, Jun 28 2011

a[1] = 1; j = 1; k = 1; n = 100; For[k = 2, k <= n, k++, If[2*a[k - j] < 3^j, a[k] = 2*a[k - j], {a[k] = 3^j, j++}]]; Table[a[i], {i, 1, n}] (* Hai He (hai(AT)mathteach.net) and Gilbert Traub, Dec 28 2004 *)
aa = {}; Do[If[EulerPhi[6 n] == 2 n, AppendTo[aa, n]], {n, 1, 1000}]; aa (* Artur Jasinski, Nov 05 2008 *)
fQ[n_] := Union[ MemberQ[{1, 5}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 6]] == {False}; fQ[1] = True; Select[ Range@ 4000, fQ] (* Robert G. Wilson v, Oct 26 2010 *)
powerOfTwo = 12; Select[Nest[Union@Join[#, 2*#, 3*#] &, {1}, powerOfTwo-1], # < 2^powerOfTwo &] (* Robert G. Wilson v and T. D. Noe, Mar 03 2011 *)
fQ[n_] := n == 3 EulerPhi@ n; Select[6 Range@ 4000, fQ]/6 (* Robert G. Wilson v, Jul 08 2011 *)
mx = 4000; Sort@ Flatten@ Table[2^i*3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}] (* Robert G. Wilson v, Aug 17 2012 *)
f[n_] := Block[{p2, p3 = 3^Range[0, Floor@ Log[3, n] + 1]}, p2 = 2^Floor[Log[2, n/p3] + 1]; Min[ Select[ p2*p3, IntegerQ]]]; NestList[f, 1, 54] (* Robert G. Wilson v, Aug 22 2012 *)
Select[Range@4000, Last@Map[First, FactorInteger@#] <= 3 &] (* Vincenzo Librandi, Aug 25 2016 *)
Select[Range[4000],Max[FactorInteger[#][[All,1]]]<4&] (* Harvey P. Dale, Jan 11 2017 *)

test(n)=for(p=2,3, while(n%p==0, n/=p)); n==1;
for(n=1,4000,if(test(n),print1(n",")))

list(lim)=my(v=List(),N);for(n=0,log(lim\1+.5)\log(3),N=3^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

is_A003586(n)=n<5||vecmax(factor(n,5)[, 1])<5 \\ M. F. Hasler, Jan 16 2015

list(lim)=my(v=List(), N); for(n=0, logint(lim\=1,3), N=3^n; while(N<=lim, listput(v, N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022

from sympy import integer_log
def A003586(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A003586gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A003586gen(), 65))) # Michael S. Branicky, Sep 17 2024
(C++) // Returns A003586 <= threshold without approximations nor sorting
#include 
std::forward_list A003586(const int threshold) {
    std::forward_list sequence;
    auto start_it = sequence.before_begin();
    for (int i = 1; i <= threshold; i *= 2) {
        for (int inc = 1; std::next(start_it) != sequence.end() && inc <= i; inc *= 3)
            ++start_it;
        auto it = start_it;
        for (int j = 1; i * j <= threshold; j *= 3) {
            sequence.emplace_after(it, i * j);
            for (int inc = 1; std::next(it) != sequence.end() && inc <= i; inc *= 2)
                ++it;
        }
    }
    return sequence;
} // Eben Gino Lester, Apr 17 2025

def isA003586(n) :
    return not any(d != 2 and d != 3 for d in prime_divisors(n))
@CachedFunction
def A003586(n) :
    if n == 1 : return 1
    k = A003586(n-1) + 1
    while not isA003586(k) : k += 1
    return k
[A003586(n) for n in (1..55)] # Peter Luschny, Jul 20 2012

An asymptotic formula for a(n) is roughly a(n) ~ 1/sqrt(6)*exp(sqrt(2*log(2)*log(3)*n)). - Benoit Cloitre, Nov 20 2001
A061987(n) = a(n + 1) - a(n), a(A084791(n)) = A084789(n), a(A084791(n) + 1) = A084790(n). - Reinhard Zumkeller, Jun 03 2003
Union of powers of 2 and 3 with n such that psi(n) = 2*n, where psi(n) = n*Product_(1 + 1/p) over all prime factors p of n = A001615(n). - Lekraj Beedassy, Sep 07 2004; corrected by Franklin T. Adams-Watters, Mar 19 2009
a(n) = 2^A022328(n)*3^A022329(n). - N. J. A. Sloane, Mar 19 2009
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} moebius(6*n)*x^n/(1 - x^n). - Paul D. Hanna, Sep 18 2011
a(n) = A007694(n+1)/2. - Lei Zhou, Apr 19 2017

Deleted claim that this sequence is union of 2^n (A000079) and 3^n (A000244) sequences -- this does not include the terms which are not pure powers. - Walter Roscello (wroscello(AT)comcast.net), Nov 16 2008

A007283 a(n) = 3*2^n.

Original entry on oeis.org

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

Non-deficient numbers (A023196) with an even sum of divisors (A000203) that are neither pseudoperfect numbers (A005835) nor Zumkeller numbers (A083207).

Equivalently, numbers k such that sigma(k) >= 2*k and sigma(k) == 0 (mod 2), such that no subset of the aliquot divisors of k sums to k or to sigma(k)/2.

Frank Buss and T. D. Noe conjectured (see A083207) and Robert Gerbicz proved that the largest possible gap between Zumkeller numbers is 12 (SeqFan post, 2010). A proof was also published by Mahanta et al. (2020).

This sequence is well defined (see A360561).
The values among the first 200000 terms are distributed as follows:
    v  # of occurrences
    -  ----------------
    1             45927
    2             45015
    3             41953
    4             18109
    5                10
    6             48986

cp's OEIS Frontend

A337399 Numbers k such that sigma(k) is a Zumkeller number (A083207).

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A337740 Weird numbers (A006037) with an even sum of divisors that are not Zumkeller numbers (A083207).

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A345704 Zumkeller numbers k (A083207) such that the next Zumkeller number is k + 12.

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A360562 a(n) is the least k such that k*n is a Zumkeller number (A083207).

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A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

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A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

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A002182 Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.