OFFSET
0,6
COMMENTS
It appears that if n > 39, then a(n) is positive, i.e., A071963(n) > n. This has been checked up to n = 2500.
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.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
J. Cilleruelo and F. Luca, On the largest prime factor of the partition function of n
A. Schinzel and E. Wirsing, Multiplicative properties of the partition function, Proc. Indian Acad. Sci., Math. Sci. (Ramanujan Birth Centenary Volume), 97 (1987), 297-303; alternative link.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Partition function
Wikipedia, Partition function
EXAMPLE
There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.
MATHEMATICA
Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]
PROG
(PARI) a(n)=if(n<2, !n, my(f=factor(numbpart(n))[, 1]); f[#f]-n) \\ Charles R Greathouse IV, Feb 04 2013
CROSSREFS
KEYWORD
sign
AUTHOR
Jonathan Sondow, Aug 16 2011
STATUS
approved