Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #26 Jun 14 2023 15:16:23
%S 1,0,0,0,1,2,5,-2,3,-4,-3,-4,-1,88,-9,-4,-5,-6,-7,-12,-1,-10,145,228,
%T -17,64,3,16,-15,54,437,280,-9,-10,1197,6,17941,244,5,-28,87,152,2375,
%U 28,53,1042,195,20,6965,582,9233,610,1,5184,5,172,963,102302
%N A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).
%C It appears that if n > 39, then a(n) is positive, i.e., A071963(n) > n. This has been checked up to n = 2500.
%C Cilleruelo and Luca proved that A071963(n) > log log n for almost all n, a much weaker statement. Earlier Schinzel and Wirsing proved that for all large N the product p(1)*p(2)*...*p(N) has at least C*log N distinct prime factors, for any positive constant C < 1/log 2.
%H T. D. Noe, <a href="/A192885/b192885.txt">Table of n, a(n) for n = 0..1000</a>
%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 A. Schinzel and E. Wirsing, <a href="http://dx.doi.org/10.1007/BF02837831">Multiplicative properties of the partition function</a>, Proc. Indian Acad. Sci., Math. Sci. (Ramanujan Birth Centenary Volume), 97 (1987), 297-303; <a href="https://www.ias.ac.in/describe/article/pmsc/097/01-03/0297-0303">alternative link</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)) - n
%e There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.
%t Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]
%o (PARI) a(n)=if(n<2,!n,my(f=factor(numbpart(n))[,1]);f[#f]-n) \\ _Charles R Greathouse IV_, Feb 04 2013
%Y Cf. A000041, A006530, A071963.
%K sign
%O 0,6
%A _Jonathan Sondow_, Aug 16 2011