A194260 A194259(n) - n, where A194259(n) is the number of distinct prime factors of p(1)*p(2)*...*p(n) and p(n) is the n-th partition number.

-1, -1, -1, -1, -1, -1, -2, -3, -4, -5, -6, -7, -7, -8, -9, -10, -11, -12, -13, -13, -14, -14, -14, -15, -15, -15, -15, -15, -15, -15, -15, -15, -16, -16, -16, -16, -16, -17, -18, -18, -18, -18, -18, -17, -17, -16, -16, -16, -16, -16, -16, -16, -16, -15, -15, -14, -14, -14, -14, -13, -13, -13, -12, -12, -12, -12, -11, -11, -10, -10, -10, -10, -9, -9, -9, -9, -9, -8, -7, -7, -7, -8, -8, -8, -8, -7, -7, -7, -7, -6, -5, -4, -4, -4, -3, -3, -4, -4, -4, -4, -4, -3, -3, -3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, 0, 1

Offset

1,7

Comments

Schinzel and Wirsing proved that a(n) > C*log n - n, for any positive constant C < 1/log 2 and all large n. In fact, it appears that a(n) > 0 for all n > 115.

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).

Links

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

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.

Formula

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

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 - 8 = -3.

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)) -n:
seq(a(n), n=1..116); # Alois P. Heinz, Aug 20 2011

Mathematica

a[n_] := PrimeNu[Product[PartitionsP[k], {k, 1, n}]] - n; Table[a[n], {n, 1, 116}] (* Jean-François Alcover, Jan 28 2014 *)

Crossrefs

Cf. A000041, A001221, A087175, A194259.

Sequence in context: A101041 A093697 A157466 * A227394 A209900 A097043

Adjacent sequences: A194257 A194258 A194259 * A194261 A194262 A194263

Keyword

sign

Author

Jonathan Sondow, Aug 20 2011

Status

approved