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

0,3

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

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.

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A087173, A192885.

Sequence in context: A381019 A381167 A087174 * A224075 A354790 A053724

Adjacent sequences: A071960 A071961 A071962 * A071964 A071965 A071966

Keyword

nonn

Author

Benoit Cloitre, Jun 16 2002

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

Status

approved