A106244 Number of partitions into distinct prime powers.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 21, 24, 27, 30, 33, 37, 41, 46, 50, 56, 62, 68, 75, 82, 91, 99, 108, 118, 129, 141, 152, 166, 180, 196, 211, 229, 248, 267, 288, 310, 335, 360, 387, 415, 447, 479, 513, 549, 589, 630, 672, 719, 768, 820, 873, 930

Offset

0,4

Comments

A054685(n) < a(n) < A023893(n) for n>2.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)

Formula

a(n) = A054685(n-1)+A054685(n). - Vladeta Jovovic, Apr 28 2005

G.f.: (1+x)*Product(Product(1+x^(p(k)^j), j=1..infinity),k=1..infinity), where p(k) is the k-th prime (offset 0). - Emeric Deutsch, Aug 27 2007

Example

a(10) = #{3^2+1,2^3+2,7+3,7+2+1,5+2^2+1,5+3+2,2^2+3+2+1} = 7.

Maple

g:=(1+x)*(product(product(1+x^(ithprime(k)^j), j=1..5), k=1..20)): gser:=series(g, x=0, 68): seq(coeff(gser, x, n), n=1..63); # Emeric Deutsch, Aug 27 2007

Mathematica

m = 64; gf = (1+x)*Product[1+x^(Prime[k]^j), {j, 1, 5}, {k, 1, 18}] + O[x]^m; CoefficientList[gf, x] (* Jean-François Alcover, Mar 02 2019, from Maple *)

Prog

(PARI) lista(m) = {x = t + t*O(t^m); gf = (1+x)*prod(k=1, m, if (isprimepower(k), (1+x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", ")); } \\ Michel Marcus, Mar 02 2019
(Haskell)
import Data.MemoCombinators (memo2, integral)
a106244 n = a106244_list !! n
a106244_list = map (p' 1) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
           where pp = a000961 k
-- Reinhard Zumkeller, Nov 24 2015

Crossrefs

Cf. A000586, A000607, A000961.

Cf. A062051, A105420, A131996.

Cf. A023893, A051613, A054685.

Sequence in context: A029024 A328188 A358637 * A029023 A140952 A096911

Adjacent sequences: A106241 A106242 A106243 * A106245 A106246 A106247

Keyword

nonn

Author

Reinhard Zumkeller, Apr 26 2005

Extensions

Offset corrected and a(0)=1 added by Reinhard Zumkeller, Nov 24 2015

Status

approved