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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. 5
-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 (list; graph; refs; listen; history; text; internal format)
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, Table of n, a(n) for n = 1..2000

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.

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-Fran├žois 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

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Last modified December 20 18:04 EST 2014. Contains 252272 sequences.