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A382983 a(n) is the number of solutions to n = x*y in positive integers x <= y where x + y is prime.

%I A382983 #12 Apr 19 2025 17:49:29
%S A382983 1,1,0,1,0,2,0,0,0,2,0,2,0,0,0,1,0,2,0,0,0,2,0,1,0,0,0,2,0,4,0,0,0,1,
%T A382983 0,2,0,0,0,2,0,4,0,0,0,1,0,1,0,0,0,2,0,1,0,0,0,2,0,4,0,0,0,0,0,2,0,0,
%U A382983 0,4,0,2,0,0,0,1,0,4,0,0,0,2,0,2,0,0,0,2
%N A382983 a(n) is the number of solutions to n = x*y in positive integers x <= y where x + y is prime.
%C A382983 a(2*k+1) = 0 for positive integers k because in x*y = 2*k + 1 >= 3 both x and y are odd, so x + y is even and > 2 and therefore not prime.
%H A382983 Felix Huber, <a href="/A382983/b382983.txt">Table of n, a(n) for n = 1..10000</a>
%F A382983 a(n) <= A038548(n).
%F A382983 a(2*k+1) = 0 for positive integers k.
%e A382983 The a(12) = 2 solutions are (1, 12) and (3, 4) because 1*12 = 12 and 1 + 12 = 13 is prime and 3*4 = 12 and 3 + 4 = 7 is prime.
%p A382983 A382983:=proc(n)
%p A382983     local a,i,L;
%p A382983     if n=1 then return 1 fi;
%p A382983     a:=0;
%p A382983     L:=NumberTheory:-Divisors(n);
%p A382983     for i to nops(L)/2 do
%p A382983         if isprime(L[i]+L[nops(L)-i+1]) then
%p A382983             a:=a+1
%p A382983         fi
%p A382983     od;
%p A382983     return a
%p A382983 end proc;
%p A382983 seq(A382983(n),n=1..88);
%o A382983 (PARI) a(n) = sumdiv(n, d, (d<=n/d) && isprime(d+n/d)); \\ _Michel Marcus_, Apr 14 2025
%Y A382983 Cf. A000040, A004526, A038548, A382984, A382985.
%K A382983 nonn,easy
%O A382983 1,6
%A A382983 _Felix Huber_, Apr 14 2025