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

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, 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, 0, 4, 0, 2, 0, 0, 0, 1, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2

Offset: 1

Author

Felix Huber, Apr 14 2025

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A004526, A038548, A382984, A382985.

Programs

Maple

A382983:=proc(n)
    local a,i,L;
    if n=1 then return 1 fi;
    a:=0;
    L:=NumberTheory:-Divisors(n);
    for i to nops(L)/2 do
        if isprime(L[i]+L[nops(L)-i+1]) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A382983(n),n=1..88);

PARI

a(n) = sumdiv(n, d, (d<=n/d) && isprime(d+n/d)); \\ Michel Marcus, Apr 14 2025

Formula

a(n) <= A038548(n).

a(2*k+1) = 0 for positive integers k.