A194259 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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)):