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A184171 Number of partitions of n into an even number of distinct primes. 10
1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 5, 5, 4, 6, 5, 5, 6, 7, 7, 8, 7, 9, 8, 9, 8, 11, 11, 12, 10, 13, 12, 14, 14, 15, 16, 17, 16, 20, 19, 20, 20, 24, 22, 26, 23, 27, 27, 30, 28, 34, 33, 36, 34, 40, 37, 43, 41, 46, 46, 50, 47, 56, 55 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,17

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Alois P. Heinz)

FORMULA

G.f.: (1/2)*[Product_{k>=1} (1+z^prime(k)) + Product_{k>=1} (1-z^prime(k))].

a(n) = Sum_{k>=0} A219180(n,2*k). - Alois P. Heinz, Nov 15 2012

EXAMPLE

a(33) = 5 because we have [31,2], [23,5,3,2], [19,7,5,2], [17,11,3,2], and [13,11,7,2].

MAPLE

g := 1/2*(Product(1+z^ithprime(k), k = 1 .. 120)+Product(1-z^ithprime(k), k = 1 .. 120)): gser := series(g, z = 0, 110): seq(coeff(gser, z, n), n = 0 .. 85);

# second Maple program

with(numtheory):

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

      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),

       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))

    end:

a:= proc(n) local l; l:= b(n, pi(n));

      add(l[2*i-1], i=1..iquo(nops(l)+1, 2))

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]}]]]; a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[2*i-1]], {i, 1, Quotient[Length[l]+1, 2]}]]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Jan 30 2014, after Alois P. Heinz *)

PROG

(PARI)

parts(n, pred, y)={prod(k=1, n, 1 + if(pred(k), y*x^k + O(x*x^n), 0))}

{my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ Andrew Howroyd, Dec 28 2017

CROSSREFS

Cf. A000586, A184172, A184198.

Sequence in context: A050252 A025877 A219182 * A133989 A029398 A025817

Adjacent sequences:  A184168 A184169 A184170 * A184172 A184173 A184174

KEYWORD

nonn

AUTHOR

Emeric Deutsch, (suggested by R. J. Mathar), Jan 09 2011

STATUS

approved

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Last modified June 21 06:04 EDT 2021. Contains 345355 sequences. (Running on oeis4.)