login
Number of partitions of n into an even number of distinct primes (counting 1 as a prime).
4

%I #6 Dec 02 2020 18:07:16

%S 1,0,0,1,1,1,1,1,2,1,1,1,2,2,2,2,3,2,3,3,4,4,4,4,6,5,5,5,6,6,7,7,9,8,

%T 9,8,11,10,11,12,14,12,15,14,17,16,17,17,22,20,22,21,25,24,28,27,31,

%U 30,33,31,39,36,40,40,46,42,49,47,54,53,58,55,67,63,70,68

%N Number of partitions of n into an even 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(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

%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), 1):

%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, A067661, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339383.

%K nonn

%O 0,9

%A _Ilya Gutkovskiy_, Dec 02 2020