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

Author

Benoit Cloitre, Jun 16 2002

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

Cf. A087173, A192885.

Programs

Mathematica

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

PARI

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

PARI

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

PARI

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

Formula

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

Extensions

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