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A184171 Number of partitions of n into an even number of distinct primes. 5
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

Alois P. Heinz, Table of n, a(n) for n = 0..2000

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 *)

CROSSREFS

Cf. A000586, A184172.

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 August 18 14:09 EDT 2017. Contains 290720 sequences.