A194259 Number of distinct prime factors of p(1)*p(2)*...*p(n), where p(n) is the n-th partition number.

0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 54, 56, 57, 59, 60, 61

Offset: 1

Author

Jonathan Sondow, Aug 20 2011

Keywords

• nonn

Comments

Schinzel and Wirsing proved that a(n) > C*log n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > n for all n > 115 (see A194260).

It also appears that a(n) > a(n-1), for all n > 97, so that some prime factor of p(n) does not divide p(1)*p(2)*...*p(n-1). See A194261, A194262.

Examples

			p(1)*p(2)*...*p(8) = 1*2*3*5*7*11*15*22 = 2^2 * 3^2 * 5^2 * 7 * 11^2, so a(8) = 5.
		

Links

• Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Alois P. Heinz)
• 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, Partition Function
• Wikipedia, Partition function

Crossrefs

Cf. A000041, A001221, A087175, A194260, A194261, A194262.

Programs

Maple

with(combinat): with(numtheory):
b:= proc(n) option remember;
      `if`(n=1, {}, b(n-1) union factorset(numbpart(n)))
    end:
a:= n-> nops(b(n)):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 20 2011

Mathematica

a[n_] := Product[PartitionsP[k], {k, 1, n}] // PrimeNu; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2014 *)
PrimeNu[FoldList[Times,PartitionsP[Range[80]]]] (* Harvey P. Dale, May 29 2025 *)

PARI

a(n)=my(v=[]);for(k=2,n,v=concat(v,factor(numbpart(k))[,1])); #vecsort(v,,8) \\ Charles R Greathouse IV, Feb 01 2013

Formula

a(n) = A001221(product(k=1..n, A000041(k))).