A024321 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (composite numbers).

0, 0, 6, 8, 9, 10, 12, 14, 25, 28, 32, 35, 37, 40, 44, 46, 64, 69, 73, 77, 81, 85, 89, 93, 96, 100, 128, 133, 139, 144, 148, 154, 162, 166, 170, 176, 181, 187, 223, 229, 236, 242, 248, 255, 262, 268, 275, 281, 287, 294, 301, 308, 354, 361, 370, 380, 386, 394, 401, 408, 418, 425

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A002808, A023531.

Programs

Magma

A002808:= [n : n in [2..100] | not IsPrime(n) ];
A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*A002808[n-j+1]: j in [1..Floor((n+1)/2)]]) : n in [1..70]]; // G. C. Greubel, Jan 19 2022

Mathematica

A023531[n_]:= SquaresR[1, 8n+9]/2;
Composite[n_]:= FixedPoint[n +PrimePi[#] +1 &, n];
a[n_]:= Sum[A023531[j]*Composite[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 70}] (* G. C. Greubel, Jan 19 2022 *)

Sage

A002808 = [n for n in (1..250) if sloane.A001222(n) > 1]
def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*A002808[n-j] for j in (1..floor((n+1)/2)) ) for n in (1..70)] # G. C. Greubel, Jan 19 2022

Formula

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A002808(n-j+1). - G. C. Greubel, Jan 19 2022