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 A184172 Number of partitions of n into an odd number of distinct primes. 10
 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 3, 5, 3, 4, 4, 5, 5, 6, 6, 7, 5, 7, 7, 8, 8, 8, 9, 11, 9, 10, 11, 12, 12, 14, 13, 16, 14, 16, 15, 19, 17, 20, 20, 22, 20, 23, 24, 27, 26, 28, 27, 33, 30, 34, 34, 37, 36, 41, 40, 46, 43, 47, 46, 55, 50, 56 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,19 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+1). - Alois P. Heinz, Nov 15 2012 EXAMPLE a(33) = 4 because we have [23,7,3], [19,11,3], [17,13,3], and [17,11,5]. 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], i=1..iquo(nops(l), 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]], {i, 1, Quotient[Length[l], 2]}]]; Table[a[n], {n, 0, 100}] (* Jean-François 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)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017 CROSSREFS Cf. A000586, A184171, A184199. Sequence in context: A306396 A105068 A120676 * A125973 A227196 A189172 Adjacent sequences:  A184169 A184170 A184171 * A184173 A184174 A184175 KEYWORD nonn AUTHOR Emeric Deutsch (suggested by R. J. Mathar), Jan 09 2011 STATUS approved

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Last modified June 23 10:51 EDT 2021. Contains 345397 sequences. (Running on oeis4.)