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

1,3

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.

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A093392 A079947 A094699 * A246264 A102676 A087846

Adjacent sequences: A194256 A194257 A194258 * A194260 A194261 A194262

Keyword

nonn

Author

Jonathan Sondow, Aug 20 2011

Status

approved