A192885 A071963(n) - n, where A071963(n) is the largest prime factor of p(n), the n-th partition number A000041(n).

1, 0, 0, 0, 1, 2, 5, -2, 3, -4, -3, -4, -1, 88, -9, -4, -5, -6, -7, -12, -1, -10, 145, 228, -17, 64, 3, 16, -15, 54, 437, 280, -9, -10, 1197, 6, 17941, 244, 5, -28, 87, 152, 2375, 28, 53, 1042, 195, 20, 6965, 582, 9233, 610, 1, 5184, 5, 172, 963, 102302

Offset: 0

Author

Jonathan Sondow, Aug 16 2011

Keywords

• sign

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.

Examples

			There are 77 partitions of 12, and 77 = 7*11, so a(12) = 11 - 12 = -1.
		

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

Crossrefs

Cf. A000041, A006530, A071963.

Programs

Mathematica

Table[First[Last[FactorInteger[PartitionsP[n]]]] - n, {n, 0, 100}]

PARI

a(n)=if(n<2,!n,my(f=factor(numbpart(n))[,1]);f[#f]-n) \\ Charles R Greathouse IV, Feb 04 2013

Formula

a(n) = A006530(A000041(n)) - n