A061742 -id:A061742 - 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

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

Original entry on oeis.org

0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853

Offset: 0

N. J. A. Sloane, Clark Kimberling

Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002
(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022
The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024
Apart from the initial 0, a subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i) [implied by Henry Bottomley's Sep 27 2006 formula], but neither does p^p divide a(i) when 0 < i < A000720(p), as then p^p > a(i). See A074107, which gives an upper bound for this sequence. - Antti Karttunen, Nov 19 2024

			0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.

Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
Index entries for sequences related to primorial numbers

Denominators are A002110.
See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.
Row sums of A077011 and A258566.
Subsequence of A048103 (after the initial 0).
Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.
Cf. A053144 (a lower bound), A074107 (an upper bound).
Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).
Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes), A377992 (antiderivatives of the terms > 1 of this sequence).
Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.

[ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ];  // Bruno Berselli, Apr 11 2011

h:= n-> add(1/(ithprime(i)),i=1..n);
t1:=[seq(h(n),n=0..50)];
t1a:=map(numer,t1); # A024451
t1b:=map(denom,t1); # A002110 - N. J. A. Sloane, Apr 25 2014

a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a,18]  (* Jean-François Alcover, Apr 11 2011 *)
f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
(* Clark Kimberling, Dec 29 2011 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)

a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018

from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021

from math import prod
from sympy import prime
def A024451(n):
    q = prod(plist:=tuple(prime(i) for i in range(1,n+1)))
    return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022

Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006
From Antti Karttunen, Jan 31 2024, Feb 08 2024 and Nov 19 2024: (Start)
a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).
For n > 0, a(n) = A327860(A143293(n-1)).
For n > 0, a(n) = A348301(n) + A002110(n).
For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]
For n >= 0, A053144(n) <= a(n) <= A074107(n) < A070826(1+n).
(End)

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

Original entry on oeis.org

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Erich Friedman

Solution for n=10 is same as for n=11.

This sequence is infinite and each term initiates a suitable arithmetic progression with large differences like squares of primorials or other suitable products of primes from prime factors being on power 2 in terms and in chains after. Proof includes solution of linear Diophantine equations and math. induction. See also A068781, A070258, A070284, A078144, A049535, A077640, A077647, A078143 of which first terms are recollected here. - Labos Elemer, Nov 25 2002

			a(3) = 48 as 48, 49 and 50 are divisible by squares.
n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the author page.
"sikefield3", double-square (2019)
Eric Weisstein's World of Mathematics, Squarefree numbers
Eric Weisstein's World of Mathematics, Squareful

Cf. A013929, A053806, A049535, A077647, A078143. Also A069021 and A051681 are different versions.

cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]

a(n)=my(s);for(k=1,9^99,if(issquarefree(k),s=0,if(s++==n,return(k-n+1)))) \\ Charles R Greathouse IV, May 29 2013

a(n) = 1 + A020754(n+1). - R. J. Mathar, Jun 25 2010
Correction from Jeppe Stig Nielsen, Mar 05 2022: (Start)
a(n) = 1 + A020754(n+1) for 1 <= n < 11.
a(n) = 1 + A020754(n) for 11 <= n < N where N is unknown.
Possibly a(n) = 1 + A020754(n-d) for some higher n, depending on how many repeated terms the sequence has. (End)
a(n) <= A061742(n) = A002110(n)^2 is the trivial bound obtained from the CRT. - Charles R Greathouse IV, Sep 06 2022

a(9)-a(11) from Patrick De Geest, Nov 15 1998, Jan 15 1999
a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999
a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al. [Louis Marmet (louis(AT)marmet.org), Jul 27 2000]
a(17) was obtained as a result of a team effort by E. Wong et al. [Louis Marmet (louis(AT)marmet.org), Jul 13 2001]
a(18) was obtained as a result of a team effort by L. Marmet et al.

A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834

Offset: 1

Erich Friedman

nonn

If a term n in the sequence ends in neither 0 nor 5, then 10*n is also in the sequence. - Lekraj Beedassy, Jun 11 2004
The lower asymptotic density of this sequence is larger than 1/18 = 0.0555... which is the density of its subsequence of numbers of the form 6*m where gcd(m, 6) = 1 and m > 1. Numerically, based on counts of terms below 10^n (A302993), it seems that this sequence has an asymptotic density which equals to about 0.070034... - Amiram Eldar, Feb 13 2021
The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - Amiram Eldar, Apr 18 2024
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, usigma(k)/k < 15/Pi^2 = 1.519817... (A082020; the record values are attained at the squares of primorials, A061742). - Amiram Eldar, Jul 20 2024

C. Sung, Mathematical Buds, "Unitary Divisors", Chap. V, pp. 42-67, Ed. H. D. Ruderman, Mu Alpha Theta OK 1978.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Subsequence of A005101.
Cf. A034444, A034448, A129487, A302993.
Cf. A001694, A061742, A082020.

isA034683 := proc(n)
    is(A034448(n) > 2*n) ;
end proc:
for n from 1 do
    if isA034683(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
Select[Range[1000], usigma[#] > 2#&] (* Jean-François Alcover, Mar 23 2020, after Giovanni Resta in A034448 *)

is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]) > 2*n;} \\ Amiram Eldar, Apr 18 2024

A162641 Number of even exponents in canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A003557, A002035, A007913, A025487, A059841, A061742 (where records occur), A072587, A162642, A179119.

Cf. A268335 (positions of zeros), A295316.

Table[Count[FactorInteger[n][[All, -1]], ?EvenQ], {n, 105}] (* _Michael De Vlieger, Jul 23 2017 *)

A162641(n) = omega(n) - omega(core(n)); \\ Antti Karttunen, Jul 23 2017

(define (A162641 n) (if (= 1 n) 0 (+ (A059841 (A067029 n)) (A162641 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = A001221(n) - A162642(n).
a(A002035(n)) = 0.
a(A072587(n)) > 0.
Additive with a(p^e) = A059841(e). - Antti Karttunen, Jul 23 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A162642(A003557(n)).
a(n) <= A056170(n).
(End)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p+1)) = 0.3302299262... (A179119). - Amiram Eldar, Dec 25 2021

A323308 The number of exponential semiproper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 10 2019

An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and gcd(d/rad(n), n/d) = 1, where rad(n) is the largest squarefree divisor of n (A007947).
a(n) is also the number of divisors of n that are squares of squarefree numbers (A062503). - Amiram Eldar, Oct 08 2022
a(n) is also the number of unitary divisors of n that are powerful (A001694). - Amiram Eldar, Feb 18 2023
The smallest integer that has exactly 2^n exponential semiproper divisors is A061742(n). - Bernard Schott, Feb 20 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7, No. 1 (2014), pp. 37-46.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to divisors of numbers.

Cf. A000188, A001694, A007947, A008833, A034444, A057521, A061742, A062503, A082020, A323309, A323310.

f[p_, e_] := If[e==1, 1, 2]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = min(f[k,2], 2); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jan 11 2019

a(n) = A034444(n/A007947(n)).
Multiplicative with a(p^e) = 1 for e = 1 and 2 otherwise.
Asymptotic mean: Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 15/Pi^2 = 1.5198177546... (A082020). - Amiram Eldar, Nov 08 2020
a(n) = Sum_{d^2|n} mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022
Dirichlet g.f.: zeta(s) * zeta(2*s) / zeta(4*s). - Werner Schulte, Dec 29 2022
a(n) = A034444(A000188(n)) = A034444(A008833(n)) (the number of unitary divisors of the largest square dividing n). - Amiram Eldar, Sep 03 2023
a(n) = A034444(A057521(n)) (the number of unitary divisors of the powerful part of n). - Amiram Eldar, Oct 03 2023

A070258 Smallest of 3 consecutive numbers each divisible by a square.

Original entry on oeis.org

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of triples of terms = {900a+548, 900a+549, 900a+550} = {4(225f+137), 9(100f+61), 25(36f+22)}; starting terms in this sequence = {548, 1448, 2348, ...}; difference = A002110(3)^2. - Labos Elemer, Nov 25 2002
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 2, 16, 180, 1868, 18649, 186335, 1863390, 18634236, 186340191, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01863... . - Amiram Eldar, Jan 18 2023
The asymptotic density of this sequence is 1 - 3/zeta(2) + 3 * Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 1 - 3 * A059956 + 3 * A065474 - A206256 = 0.018634010349844827414... . - Amiram Eldar, Sep 12 2024

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A013929 and A068781.
Cf. A002110, A061742, A235578.
Cf. A059956, A065474, A206256.

f[n_] := Union[ Transpose[ FactorInteger[n]] [[2]]] [[ -1]]; a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, Print[n - 2]]; a = b; b = c, {n, 3, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[3000],3,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
f@n_ := Flatten@  Position[Partition[SquareFreeQ /@ Range@2000, n, 1], Table[False, {n}]]; f@3 (* Hans Rudolf Widmer, Aug 30 2022 *)

a(n) = A235578(n) - 1. - Amiram Eldar, Feb 09 2021

More terms from Jason Earls and Robert G. Wilson v, May 10 2002

Offset corrected by Amiram Eldar, Feb 09 2021

A166469 Number of divisors of n which are not multiples of consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 3, 5, 2, 4, 2, 6, 4, 4, 2, 5, 3, 4, 4, 6, 2, 5, 2, 6, 4, 4, 3, 5, 2, 4, 4, 8, 2, 6, 2, 6, 4, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 8, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 6, 2, 6, 2, 4, 4, 6, 3, 6, 2, 10, 5, 4, 2, 8, 4, 4, 4, 8, 2, 6, 4, 6, 4, 4, 4, 7, 2, 6, 6, 9, 2, 6, 2, 8, 5

Offset: 1

Matthew Vandermast, Nov 05 2009

nonn

Links various subsequences of A025487 with an unusual number of important sequences, including the Fibonacci, Lucas, and other generalized Fibonacci sequences (see cross-references).

If a number is a product of any number of consecutive primes, the number of its divisors which are not multiples of n consecutive primes is always a Fibonacci n-step number. See also A073485, A167447.

			Since 3 of 30's 8 divisors (6, 15, and 30) are multiples of 2 or more consecutive primes, a(30) = 8 - 3 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A000040, A006094, A104210, A296210.
A(A002110(n)) = A000045(n+2); A(A097250(n)) = A000032(n+1). For more relationships involving Fibonacci and Lucas numbers, see A166470-A166473, comment on A081341.
A(A061742(n)) = A001045(n+2); A(A006939(n)) = A000085(n+1); A(A212170(n)) = A000142(n+1). A(A066120(n)) = A166474(n+1).

Array[DivisorSum[#, 1 &, FreeQ[Differences@ PrimePi@ FactorInteger[#][[All, 1]], 1] &] &, 105] (* Michael De Vlieger, Dec 16 2017 *)

A296210(n) = { if(1==n,return(0)); my(ps=factor(n)[,1], pis=vector(length(ps),i,primepi(ps[i])), diffsminusones = vector(length(pis)-1,i,(pis[i+1]-pis[i])-1)); !factorback(diffsminusones); };
A166469(n) = sumdiv(n,d,!A296210(d)); \\ Antti Karttunen, Dec 15 2017

a) If n has no prime gaps in its factorization (cf. A073491), then, if the canonical factorization of n into prime powers is the product of p_i^(e_i), a(n) is the sum of all products of one or more nonadjacent exponents, plus 1. For example, if A001221(n) = 3, a(n) = e_1*e_3 + e_1 + e_2 + e_3 + 1. If A001221(n) = k, the total number of terms always equals A000045(k+2).
The answer can also be computed in k steps, by finding the answers for the products of the first i powers, for i = 1 to i = k. Let the result of the i-th step be called r(i). r(1) = e_1 + 1; r(2) = e_1 + e_2 +1; for i > 2, r(i) = r(i-1) + e_i * r(i-2).
b) If n has prime gaps in its factorization, express it as a product of the minimum number of A073491's members possible. Then apply either of the above methods to each of those members, and multiply the results to get a(n). a(n) = A000005(n) iff n has no pair of consecutive primes as divisors.
a(n) = Sum_{d|n} (1-A296210(d)). - Antti Karttunen, Dec 15 2017

Edited by Matthew Vandermast, May 24 2012

cp's OEIS Frontend

A189997 Partial sums of A061742.

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

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A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

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A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).

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A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

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