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A051613 a(n) = partitions of n into powers of distinct primes (1 not considered a power). 12
1, 0, 1, 1, 1, 2, 0, 3, 2, 3, 2, 4, 3, 4, 4, 4, 8, 4, 8, 6, 9, 8, 10, 10, 13, 12, 13, 16, 16, 19, 17, 21, 23, 23, 25, 29, 31, 31, 31, 37, 40, 42, 44, 48, 49, 54, 55, 64, 67, 68, 70, 77, 84, 90, 92, 99, 102, 108, 115, 127, 133, 135, 138, 150, 165, 171, 183, 186, 198, 201, 220 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

REFERENCES

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = number of m such that A008475(m) = n.

G.f.: Product_{p prime} (1 + Sum_{k >= 1} x^(p^k)).

EXAMPLE

a(16) = 8 because we can write 16 = 2^4 = 3+13 = 5+11 = 3^2+7 = 2+3+11 = 2+3^2+5 = 2^3+3+5 = 2^2+5+7.

MAPLE

b:= proc(n, i) option remember; local p;

      p:= `if`(i<1, 1, ithprime(i));

      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+

      add(b(n-p^j, i-1), j=1..ilog[p](n))))

    end:

a:= n-> b(n, numtheory[pi](n)):

seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

MATHEMATICA

max = 70; f[x_] := Product[ 1 + Sum[x^(Prime[n]^k), {k, 1, If[n > 4, 1, 6]}], {n, 1, PrimePi[max]}]; CoefficientList[ Series[f[x], {x, 0, max}] , x](* Jean-Fran├žois Alcover, Sep 12 2012 *)

PROG

(Haskell)

import Data.MemoCombinators (memo3, integral)

a051613' = p 1 2 where

   p x _ 0 = 1

   p x k m | m < qq       = 0

           | mod x q == 0 = p x (k + 1) m

           | otherwise    = p (q * x) (k + 1) (m - qq) + p x (k + 1) m

           where q = a025473 k; qq = a000961 k

-- Reinhard Zumkeller, Nov 23 2015

CROSSREFS

Cf. A023894, A009490, A054685, A008475.

Cf. A106245.

Cf. A000961, A025473, A106244.

Sequence in context: A132623 A277890 A243403 * A173291 A077961 A077962

Adjacent sequences:  A051610 A051611 A051612 * A051614 A051615 A051616

KEYWORD

nonn,nice,easy

AUTHOR

Vladeta Jovovic

EXTENSIONS

Better description from David W. Wilson, Apr 19 2000

STATUS

approved

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Last modified December 2 21:20 EST 2016. Contains 278694 sequences.