%I #5 Dec 02 2020 18:08:52
%S 0,1,1,1,0,1,1,1,1,1,2,2,1,2,2,2,2,3,3,4,3,4,4,5,4,5,6,6,5,7,6,8,7,8,
%T 9,10,9,12,11,12,11,14,14,16,15,17,17,20,17,21,22,24,22,27,25,30,28,
%U 31,31,36,33,40,39,42,40,47,46,53,49,55,54,63,58,68,67,73
%N Number of partitions of n into an odd number of distinct primes (counting 1 as a prime).
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) - (1 - x) * Product_{k>=1} (1 - x^prime(k))).
%F a(n) = (A036497(n) - A298602(n)) / 2.
%e a(21) = 4 because we have [17, 3, 1], [13, 7, 1], [13, 5, 3] and [11, 7, 3].
%p s:= proc(n) option remember;
%p `if`(n<1, n+1, ithprime(n)+s(n-1))
%p end:
%p b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
%p `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
%p b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
%p end:
%p a:= n-> b(n, numtheory[pi](n), 0):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Dec 02 2020
%t nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] - (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y Cf. A000586, A008578, A036497, A067659, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339382.
%K nonn
%O 0,11
%A _Ilya Gutkovskiy_, Dec 02 2020
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