A024320 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 = (1, p(1), p(2), ... ).

0, 0, 2, 3, 5, 7, 11, 13, 24, 30, 36, 46, 50, 60, 70, 74, 103, 117, 131, 139, 157, 171, 177, 193, 207, 221, 278, 294, 310, 330, 348, 360, 390, 408, 424, 448, 470, 486, 573, 611, 625, 653, 673, 699, 739, 761, 781, 803, 835, 863, 891, 925, 1054, 1078, 1104, 1136, 1180, 1214

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, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A000040, A023531.

Programs

Magma

A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
p:= func< n | n eq 1 select 1 else NthPrime(n-1) >;
[ (&+[A023531(j)*p(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..60]]; // G. C. Greubel, Jan 19 2022

Mathematica

A023531[n_]:= SquaresR[1, 8*n+9]/2;
p[n_]:= If[n==1, 1, Prime[n-1]];
a[n_]:= Sum[A023531[j]*p[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 60}] (* G. C. Greubel, Jan 19 2022 *)

Sage

def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
def p(n):
    if (n==1): return 1
    else: return nth_prime(n-1)
[sum( A023531(j)*p(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..60)] # G. C. Greubel, Jan 19 2022

Formula

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