A024937 a(n) = number of 2's in all partitions of n into distinct primes.

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3, 5, 4, 4, 5, 5, 6, 6, 5, 7, 7, 7, 8, 8, 9, 8, 9, 11, 11, 10, 12, 12, 13, 14, 14, 16, 15, 16, 17, 19, 20, 20, 20, 22, 24, 23, 26, 27, 27, 28, 30, 33, 34, 34, 36, 37, 40, 41, 43, 46, 46, 47, 50, 55, 56, 56, 58, 63, 64

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A024939.

Programs

Maple

with(numtheory):
b:= proc(n, i) option remember; local g;
    if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i-1));
         b(n, i-1) +g +[0, `if`(i=1,g[1],0)]
    fi
end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=0..80);  # Sean A. Irvine, after Alois P. Heinz, Jul 29 2019

Mathematica

max = 100; (* number of terms *)
CoefficientList[x^2*Product[1+x^Prime[k], {k, 2, PrimePi[max]}]+O[x]^max, x] (* Jean-François Alcover, Sep 07 2022, after Vladeta Jovovic *)

Formula

G.f.: x^2*Product_{k>1} (1+x^prime(k)). - Vladeta Jovovic, Jul 20 2003

Extensions

More terms from Vladeta Jovovic, Jul 20 2003

a(0)-a(6) prepended by Sean A. Irvine, Jul 29 2019