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A106244
Number of partitions into distinct prime powers.
27
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 26 2005
EXTENSIONS
Offset corrected and a(0)=1 added by Reinhard Zumkeller, Nov 24 2015
STATUS
approved