A036691 -id:A036691 - cp's OEIS frontend

A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).

1, 4, 6, 4, 10, 6, 14, 6, 9, 10, 22, 6, 26, 14, 10, 8, 34, 9, 38, 10, 14, 22, 46, 6, 15, 26, 9, 14, 58, 10, 62, 8, 22, 34, 14, 9, 74, 38, 26, 10, 82, 14, 86, 22, 10, 46, 94, 8, 21, 15, 34, 26, 106, 9, 22, 14, 38, 58

Offset: 1

Lechoslaw Ratajczak, Mar 25 2022

nonn

a(n) >= 2*gpf(n) for n > 1, where gpf(n) denotes the greatest prime factor of n (A006530(n)).
Conjecture: the equation a(n) = a(n+1) has no solutions. This holds up to at least n = 10^7.
Consecutive solutions of the equation a(n) = 2*K(n) (where K(n) is the Kempner number A002034(n)) are consecutive terms of A048839.

			a(15) = 10 because:
15 does not divide 1=A036691(0)=1, 1*4=A036691(1)=4, 1*4*6=A036691(2)=24, 1*4*6*8=A036691(3)=192, 1*4*6*8*9=A036691(4)=1728 and does divide 1*4*6*8*9*10=A036691(5)=17280.

J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Cf. A002034, A006530, A018252, A036691.

f(p,k):=(z:2, for m:2 while -1+sum(floor((p*m)/(p^t)),t,1,m)

a(p) = 2*p for prime p.
a(p_1*p_2*...*p_u) = 2*p_u, where p_i's are distinct primes and p_1 < p_2 < ... < p_u.
a(n) where n is factored as n = p_1^k_1*p_2^k_2*...*p_u^k_u is given by a(n) = max( a(p_1^k_1), a(p_2^k_2), ..., a(p_u^k_u) ), where a(p_i^k_i) = w*p_i and w is the smallest m >= 2 satisfying the inequality:
-1 + Sum_{t=1..m} floor((m*p_i)/(p_i)^t) >= k_i.

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

Original entry on oeis.org

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

A049614 n! divided by its squarefree kernel.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Labos Elemer

nonn

Also product of composite numbers less than or equal to n. - Benoit Cloitre, Aug 18 2002
Also n! divided by n primorial (or n!/n#). - Cino Hilliard, Mar 26 2006
From Alexander R. Povolotsky and Peter J. C. Moses, Aug 27 2007: (Start)
It appears that a(n) = smallest positive number m such that the sequence b(n) = { m (i^1 + 1!) (i^2 + 2!) ... (i^n + n!) / n! : i >= 0 } takes integral values. [It would be nice to have a proof of this! - N. J. A. Sloane] Cf. A064808 (for n=2), A131682 (for n=3), A131683 (for n=4), A131527 (for n=5), A131684 (for n=6), A131528. See also A129995, A131685. (End)
It appears that every term > 4 is divisible by 24. - Alexander R. Povolotsky, Oct 18 2007
The above comment is correct since each term divides the next. - Charles R Greathouse IV, Jan 16 2012
When n is not a prime number, then a(n)=m*n, where m is some integer >0; such a(n) make up the A036691 Otherwise, when n is a prime number, then a(n)=a(k), where k is the largest nonprime number preceding n (kAlexander R. Povolotsky, Aug 21 2012

			n = 11: 11! = 39916800 = 2310*17280 and 2310=2*3*5*7*11.

Cf. A000142, A002110, A003418, A034386, A036691, A045948, A048148.

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
[A049614(n): n in [0..40]]; // G. C. Greubel, Jul 21 2023

primorial := n -> mul(k, k=select(isprime, [$1..n]));
A049614 := n -> factorial(n)/primorial(n);
seq(A049614(i),i=0..24); # Peter Luschny, Feb 16 2013

Table[n!/Product[ Prime[i], {i, PrimePi[n]}], {n, 24}]

PARI
```
a(n)=prod(i=1,n,i^if(isprime(i),0,1))
```

a(n)=n!/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Aug 30 2012

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
[A049614(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

a(n) = A000142(n)/A034386(n).

Edited by N. J. A. Sloane, Oct 07 2007

Offset set to 0, a(0)=1 prepended to data, Peter Luschny, Feb 16 2013

A053767 Sum of first n composite numbers.

Original entry on oeis.org

0, 4, 10, 18, 27, 37, 49, 63, 78, 94, 112, 132, 153, 175, 199, 224, 250, 277, 305, 335, 367, 400, 434, 469, 505, 543, 582, 622, 664, 708, 753, 799, 847, 896, 946, 997, 1049, 1103, 1158, 1214, 1271, 1329, 1389, 1451, 1514, 1578, 1643, 1709, 1777, 1846, 1916, 1988

Offset: 0

G. L. Honaker, Jr., Mar 29 2000

nonn

a(A196415(n)) = A036691(A196415(n)) / A141092(n). - Reinhard Zumkeller, Oct 03 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Rivera, See also

Cf. A002808, A051349.

First differences of A023539.

a053767 n = a053767_list !! (n-1)
a053767_list = scanl1 (+) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A053767 := proc(n)
        add(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011
ListTools[PartialSums](remove(isprime,[$2..1000])); # Robert Israel, Jan 09 2015

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 2, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Complement[Range[2,100],Prime[Range[PrimePi[100]]]]] (* Harvey P. Dale, Dec 28 2010 *)
Accumulate[Select[Range[2, 100], ! PrimeQ[#] &]]

lista(nn) = {my(s=0); forcomposite(n=0, nn, print1(s, ", "); s += n;);} \\ Michel Marcus, Jan 09 2015

a(n) = A000217(A002808(n)) - A034387(A002808(n)) - 1 . - Robert Israel, Jan 09 2015

a(n) = A051349(n+1) - 1. - Michel Marcus, Feb 16 2018

a(0)=0 prepended by Max Alekseyev, Feb 10 2018

A141092 Product of first k composite numbers divided by their sum, when the result is an integer.

Original entry on oeis.org

1, 64, 46080, 111974400, 662171811840, 310393036800000, 7230916185292800, 108238138194410864640000, 23835710455777670400935290994688000000000, 1104077556971139123493322971152384000000000

Offset: 1

Enoch Haga, Jun 01 2008

Find the products and sums of first k composites, k = 1, 2, 3, .... When the products divided by the sums produce integral quotients, add terms to sequence.

			a(3)=46080 because 4*6*8*9*10*12*14=2903040 and 4+6+8+9+10+12+14=63; 2903040/63=46080, which is an integer, so 46080 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..90

Cf. A196415, A141089, A141090, A141091.
Compare with A140761, A159578, A140763, A116536.
Cf. A116536.

import Data.Maybe (catMaybes)
a141092 n = a141092_list !! (n-1)
a141092_list = catMaybes $ zipWith div' a036691_list a053767_list where
   div' x y | m == 0    = Just x'
            | otherwise = Nothing where (x',m) = divMod x y
-- Reinhard Zumkeller, Oct 03 2011

With[{cnos=Select[Range[50],CompositeQ]},Select[Table[Fold[ Times,1,Take[ cnos,n]]/ Total[Take[cnos,n]],{n,Length[cnos]}],IntegerQ]] (* Harvey P. Dale, Jan 14 2015 *)

s=0;p=1;forcomposite(n=4,100,p*=n;s+=n;if(p%s==0,print1(p/s", "))) \\ Charles R Greathouse IV, Apr 04 2013

a(n) = A036691(A196415(n)) / A053767(A196415(n)). [Reinhard Zumkeller, Oct 03 2011]

Checked by N. J. A. Sloane, Oct 02 2011.

A092435 Prime factorials divided by their corresponding primorials.

Original entry on oeis.org

1, 1, 4, 24, 17280, 207360, 696729600, 12541132800, 115880067072000, 1366643159020339200000, 40999294770610176000000, 1854768736099424576471040000000, 109950690675973888893203251200000000, 4617929008390903333514536550400000000, 420600974084243475616503989010432000000000

Offset: 1

Don Willard (dwillard(AT)prairie.cc.il.us), Mar 23 2004

nonn

			E.g., 2 factorial divided by 2 primorial is 1; 3 factorial is 6, divided by 3 primorial (3*2=6) is also 1; 5 factorial is 120, divided by 5 primorial (5*3*2=30) is 4 and so forth.

Subsequence of A036691. - Chayim Lowen, Jul 23 2015

Cf. A002110, A039716.

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1)*mul(i, i=ithprime(n-1)+1..ithprime(n)-1))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Jan 15 2025

Table[ Prime[n]! / Times @@ Prime[ Range[ n]], {n, 13}] (* Robert G. Wilson v, Mar 25 2004 *)

a(n)=prime(n)!/prod(i=1,n,prime(i)) \\ Ralf Stephan, Dec 21 2013

p!/p# = A039716/A002110.
Partial products of A061214. - Lekraj Beedassy, Nov 06 2006
From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A036691(A065890(n)).
a(n) = A000142(A002808(A065890(n)))/A034386(A002808(A065890(n))).
a(n) = Product_{k=1..n} prime(k)^(A085604(prime(n),k)-1).
a(n) = A049614(prime(n)).
a(n) = Product_{k=1..prime(n)} k^A066247(k). (End)

Edited by Robert G. Wilson v, Mar 25 2004

More terms from Michel Marcus, Jan 15 2025

A196415 Values of n such that (product of first n composite numbers) / (sum of first n composite numbers) is an integer.

Original entry on oeis.org

1, 4, 7, 10, 13, 15, 16, 21, 32, 33, 56, 57, 60, 70, 77, 80, 83, 84, 88, 92, 93, 97, 112, 114, 115, 120, 122, 130, 134, 141, 147, 153, 155, 164, 165, 188, 191, 196, 201, 202, 213, 222, 225, 226, 229, 243, 245, 248, 252, 260, 264, 265, 268, 273, 274, 281

Offset: 1

N. J. A. Sloane, Oct 02 2011

nonn

A036691(a(n)) mod A053767(a(n)) = 0, A141092(n) = A036691(a(n)) / A053767(a(n)). [Reinhard Zumkeller, Oct 03 2011]

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A051838, A141090, A141091, A141092.

import Data.List (elemIndices)
a196415 n = a196415_list !! (n-1)
a196415_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod a036691_list a053767_list
-- Reinhard Zumkeller, Oct 03 2011

# First define list of composite numbers:
tc:=[4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,
28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,
50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,
70,72,74,75,76,77,78,80,81,82,84,85,86,87,88];
a1:=n->mul(tc[i],i=1..n);
a2:=n->add(tc[i],i=1..n);
sn:=[];
s0:=[];
s1:=[];
s2:=[];
for n from 1 to 40 do
  t1:=a1(n)/a2(n);
  if whattype(t1) = integer then
   sn:= [op(sn),n];
   s0:= [op(s0),t1];
   s1:= [op(s1),a1(n)];
   s2:= [op(s2),a2(n)];
fi;
od:
sn; s0; s1; s2;
# alternatively
for n from 1 to 1000 do
        if type(A036691(n)/A053767(n),'integer') then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Oct 03 2011

c = Select[Range[2,355], ! PrimeQ@# &]; p = 1; s = 0; Select[Range@ Length@c, Mod[p *= c[[#]], s += c[[#]]] == 0 &] (* Giovanni Resta, Apr 03 2013 *)

More terms from Arkadiusz Wesolowski, Oct 03 2011

A025543 Least common multiple of the first n composite numbers.

Original entry on oeis.org

1, 4, 12, 24, 72, 360, 360, 2520, 2520, 5040, 5040, 5040, 5040, 55440, 55440, 277200, 3603600, 10810800, 10810800, 10810800, 21621600, 21621600, 367567200, 367567200, 367567200, 6983776800, 6983776800, 6983776800, 6983776800, 6983776800

Offset: 0

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index to divisibility sequences

Differs from A003418 and A051451
Cf. A002808, A003418, A051451, A064354.
Cf. A036691.

a025543 n = a025543_list !! n
a025543_list = scanl lcm 1 a002808_list
-- Reinhard Zumkeller, Nov 10 2013

Table[Apply[LCM, Take[Select[Range[2, 300], !PrimeQ[#] &], n]], {n, 1, 100}]  (* Clark Kimberling, Nov 12 2016 *)

A065897 The a(n)-th composite number is twice the n-th prime.

Original entry on oeis.org

1, 2, 5, 7, 13, 16, 22, 25, 31, 41, 43, 52, 59, 62, 69, 78, 87, 91, 101, 107, 111, 120, 127, 137, 149, 155, 159, 166, 170, 177, 199, 206, 215, 218, 235, 239, 248, 259, 266, 277, 286, 289, 306, 309, 316, 319, 339, 359, 366, 369, 375, 386, 389, 406, 416, 426, 438

Offset: 1

Labos Elemer, Nov 28 2001

Also the least k such that the n-th primorial (A002110) is a divisor of the k-th compositorial (A036691). - Reinhard Zumkeller, Sep 03 2002

			a(7) = 22 because twice the 7th prime (2*17 = 34) is the 22nd composite number: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000720, A002110, A002808, A036691, A100484 (even semiprimes).

A065897:= func< n | 2*NthPrime(n) -1 -#PrimesUpTo(2*NthPrime(n)) >;
[A065897(n): n in [1..130]]; // G. C. Greubel, Aug 24 2024

A065897:=n->2*ithprime(n)-(numtheory[pi](2*ithprime(n)))-1: seq(A065897(n), n=1..100); # Wesley Ivan Hurt, Sep 16 2017

Table[2*Prime[n]-(PrimePi[2*Prime[n]])-1, {n, 128}]

{ for (n=1, 1000, f=2*prime(n); a=f - primepi(f) - 1; write("b065897.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 04 2009

def A065897(n): return 2*nth_prime(n) -prime_pi(2*nth_prime(n)) -1
[A065897(n) for n in range(1,131)] # G. C. Greubel, Aug 24 2024

a(n) = 2*prime(n) - (pi(2*prime(n))) - 1, where pi = A000720.

A196529 Half of greatest common divisor of products of first n prime numbers and first n composite numbers.

Original entry on oeis.org

1, 3, 3, 3, 15, 15, 105, 105, 105, 105, 105, 105, 1155, 1155, 1155, 15015, 15015, 15015, 15015, 15015, 15015, 255255, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 4849845, 4849845, 111546435, 111546435, 111546435, 111546435, 111546435, 111546435

Offset: 1

Reinhard Zumkeller, Oct 03 2011

nonn

a(n) = gcd(A002110(n),A036691(n)) / 2.

			a(3) = gcd(2*3*5,4*6*8)/2 = gcd(30,192)/2 = 6/2 = 3;
a(4) = gcd(2*3*5*7,4*6*8*9)/2 = gcd(210,1728)/2 = 6/2 = 3;
a(5) = gcd(2*3*5*7*11,4*6*8*9*10)/2 = gcd(2310,17280)/2 = 30/2 = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A196527, A070826.

nn=40;With[{prs=Prime[Range[nn]],comps=Take[Complement[Range[Prime[nn]], Prime[ Range[nn]]],nn]},Rest[Table[GCD[Times@@Take[prs,n], Times@@Take[ comps,n]]/2,{n,nn}]]] (* Harvey P. Dale, Oct 16 2011 *)

cp's OEIS Frontend

A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Sage

Scheme

Formula

A049614 n! divided by its squarefree kernel.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A053767 Sum of first n composite numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A141092 Product of first k composite numbers divided by their sum, when the result is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A092435 Prime factorials divided by their corresponding primorials.