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A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).

%I A341973 #15 Oct 01 2021 09:46:05
%S A341973 1,1,1,1,1,2,1,1,0,2,1,2,1,2,0,3,1,3,1,2,0,4,1,2,0,2,0,4,1,3,1,3,0,4,
%T A341973 0,2,1,3,0,5,1,4,1,3,0,6,1,4,0,3,0,6,1,3,0,3,0,7,1,3,1,5,0,6,0,3,1,5,
%U A341973 0,7,1,5,1,5,0,7,0,5,1,4,0,9,1,4,0,4,0,10,1,4,0,4,0,7
%N A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).
%H A341973 Alois P. Heinz, <a href="/A341973/b341973.txt">Table of n, a(n) for n = 3..10000</a>
%F A341973 a(n) = A117929(n) + A010051(n-1). - _R. J. Mathar_, Oct 01 2021
%p A341973 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A341973      `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
%p A341973      `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 3)
%p A341973     end:
%p A341973 a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
%p A341973 seq(a(n), n=3..96);  # _Alois P. Heinz_, Feb 24 2021
%t A341973 a[n_] := Select[IntegerPartitions[n, {2}, Join[{1},
%t A341973     Prime[Range[PrimePi[n-1]]]]], #[[1]] != #[[2]]&] // Length;
%t A341973 a /@ Range[3, 100] (* _Jean-François Alcover_, Jul 13 2021 *)
%Y A341973 Cf. A008578, A036497, A117929, A154804, A341945, A341974, A341975, A341976, A341977.
%K A341973 nonn
%O A341973 3,6
%A A341973 _Ilya Gutkovskiy_, Feb 24 2021