%I #35 Oct 30 2023 08:15:37
%S 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,
%T 29,43,13,83,467,311,23,23,1231,41,17977,281,43,11,127,193,2417,71,97,
%U 1087,241,67,7013,631,9283,661,53,5237,59,227,1019,102359,3251,199,409,971
%N Largest prime factor of p(n), the n-th partition number A000041(n) (with a(0) = a(1) = 1 by convention).
%C Cilleruelo and Luca prove that a(n) > log log n, for almost all n.
%C 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
%H Giovanni Resta, <a href="/A071963/b071963.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H J. Cilleruelo and F. Luca, <a href="http://www.uam.es/personal_pdi/ciencias/cillerue/Papers/CLPofpofnAA.pdf">On the largest prime factor of the partition function of n</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Partition_(number_theory)">Partition function</a>.
%F a(n) = A006530(A000041(n)).
%e A000041(110) = 607163746 = 2*7*4049*10711, therefore a(110)=10711. - _Reinhard Zumkeller_, Aug 23 2003
%t Table[First[Last[FactorInteger[PartitionsP[n]]]], {n, 0, 100}] (* Jonathan Sondow, Aug 16 2011 *)
%o (PARI) for(n=2,75,print1(vecmax(component(factor(polcoeff(1/eta(x),n,x)),1)),","))
%o (PARI) a(n)=local(v); if(n<2,n>=0,v=factor(polcoeff(1/eta(x+x*O(x^n)),n))~[1,]; v[ #v])
%o (PARI) a(n)=if(n<2,1,factor(numbpart(n))[1,1]) \\ _Charles R Greathouse IV_, May 29 2015
%Y Cf. A087173, A192885.
%K nonn
%O 0,3
%A _Benoit Cloitre_, Jun 16 2002
%E Corrected by _T. D. Noe_, Nov 15 2006
%E Edited by _N. J. A. Sloane_, Oct 27 2008 at the suggestion of _R. J. Mathar_
%E a(0) = 1 added by _N. J. A. Sloane_, Sep 13 2009