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
Keywords
Examples
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.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
Crossrefs
Programs
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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 -
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
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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 *) -
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
Formula
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
Extensions
Offset corrected and a(0)=1 added by Reinhard Zumkeller, Nov 24 2015
Comments