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A111901 Number of partitions of n into parts that are primes or squares of primes. 5
1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 30, 37, 44, 53, 63, 75, 89, 105, 123, 145, 169, 197, 229, 266, 307, 355, 408, 469, 538, 615, 703, 801, 912, 1035, 1175, 1330, 1504, 1698, 1914, 2155, 2423, 2721, 3051, 3418, 3824, 4273, 4770, 5319, 5925 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: Product_{k>=1} 1/((1 - x^prime(k))*(1 - x^(prime(k)^2))). - Ilya Gutkovskiy, Dec 26 2016

EXAMPLE

G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 + ...

a(10) = #{ 7+3, 5+5, 5+3+2, 2^2+2^2+2, 2^2+3+3, 2^2+2+2+2, 3+3+2+2, 2+2+2+2+2 } = 8.

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

     `if`(tau(d) in [2, 3], d, 0), d=divisors(j)), j=1..n)/n)

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Mar 30 2017

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*DivisorSum[j, If[2 <= DivisorSigma[0, #] <= 3, #, 0]&], {j, 1, n}]/n];

Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Apr 06 2017, after Alois P. Heinz *)

PROG

(PARI) {a(n) = if(n < 0, 0, polcoeff( 1 / prod(k=1, primepi(n), (1 - x^prime(k)^2 + x*O(x^n)) * (1 - x^prime(k))), n))}; /* Michael Somos, Dec 26 2016 */

CROSSREFS

Cf. A000607, A090677, A023893, A111902.

Sequence in context: A095699 A112192 A017841 * A134727 A152305 A131419

Adjacent sequences:  A111898 A111899 A111900 * A111902 A111903 A111904

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Aug 20 2005

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.