A087173 -id:A087173 - cp's OEIS frontend

A071963 Largest prime factor of p(n), the n-th partition number A000041(n) (with a(0) = a(1) = 1 by convention).

1, 1, 2, 3, 5, 7, 11, 5, 11, 5, 7, 7, 11, 101, 5, 11, 11, 11, 11, 7, 19, 11, 167, 251, 7, 89, 29, 43, 13, 83, 467, 311, 23, 23, 1231, 41, 17977, 281, 43, 11, 127, 193, 2417, 71, 97, 1087, 241, 67, 7013, 631, 9283, 661, 53, 5237, 59, 227, 1019, 102359, 3251, 199, 409, 971

Offset: 0

Benoit Cloitre, Jun 16 2002

nonn

Cilleruelo and Luca prove that a(n) > log log n, for almost all n.

By computation, a(n) > log n, at least up to n = 2500. In fact, a(n) > n if n > 39, at least up to n = 2500; see A192885. - Jonathan Sondow, Aug 16 2011

			A000041(110) = 607163746 = 2*7*4049*10711, therefore a(110)=10711. - _Reinhard Zumkeller_, Aug 23 2003

Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
J. Cilleruelo and F. Luca, On the largest prime factor of the partition function of n.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Eric Weisstein's World of Mathematics, Partition Function.
Wikipedia, Partition function.

Cf. A087173, A192885.

Table[First[Last[FactorInteger[PartitionsP[n]]]], {n, 0, 100}] (* Jonathan Sondow, Aug 16 2011 *)

for(n=2,75,print1(vecmax(component(factor(polcoeff(1/eta(x),n,x)),1)),","))

a(n)=local(v); if(n<2,n>=0,v=factor(polcoeff(1/eta(x+x*O(x^n)),n))~[1,]; v[ #v])

a(n)=if(n<2,1,factor(numbpart(n))[1,1]) \\ Charles R Greathouse IV, May 29 2015

a(n) = A006530(A000041(n)).

Corrected by T. D. Noe, Nov 15 2006
Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar
a(0) = 1 added by N. J. A. Sloane, Sep 13 2009

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.

Original entry on oeis.org

1, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 33, 38, 39, 82, 97, 158, 166, 180

Offset: 1

Jonathan Sondow, Aug 21 2011

It appears that for all k > 180, the largest prime factor of p(k) does not divide p(1)*p(2)*...*p(k-1). This has been checked up to k = 2000. [Checked up to k = 10000, using A071963 b-file. - Pontus von Brömssen, Jun 05 2023]

See A071963 and A194259 for links and additional comments.

			1 is in the sequence because p(1) = 1 and 1 has no prime factor, so the condition is vacuously true.
For k = 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 33, 38, 39, 82, 97, every prime factor of p(k) divides p(1)*p(2)*...*p(k-1).
For k = 158, 166, 180, not every prime factor of p(k) divides p(1)*p(2)*...*p(k-1), but the largest one does.

Cf. A000041, A071963, A087173, A192885, A194259, A194260, A194261, A194262.

cp's OEIS Frontend

A071963 Largest prime factor of p(n), the n-th partition number A000041(n) (with a(0) = a(1) = 1 by convention).

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A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.

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cp's OEIS Frontend

A071963 Largest prime factor of p(n), the n-th partition number A000041(n) (with a(0) = a(1) = 1 by convention).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)*p(2)*...*p(k-1), where p(k) is the number of partitions of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A194345 Numbers k for which the largest prime factor of p(k) divides p(1)p(2)...*p(k-1), where p(k) is the number of partitions of k.