A023672 Convolution of A023533 and primes.

2, 3, 5, 9, 14, 18, 24, 30, 36, 48, 53, 65, 77, 85, 97, 111, 121, 131, 149, 163, 174, 192, 204, 220, 242, 260, 272, 294, 310, 320, 350, 364, 382, 410, 436, 453, 469, 495, 513, 543, 569, 587, 615, 647, 661, 687, 715, 739, 759, 799, 827, 855, 869

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000040, A023533.

Programs

Magma

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[NthPrime(k)*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 18 2022

Mathematica

A023672[n_, m_]:= A023672[n, m]= Sum[Prime[(m +Binomial[n+2,3] -Binomial[j+2, 3])], {j, n}];
Table[A023672[n, m], {n,10}, {m,Binomial[n+2,2]}]//Flatten (* G. C. Greubel, Jul 18 2022 *)

SageMath

def A023672(n,k): return sum(nth_prime(k +binomial(n+2,3) -binomial(j+2,3)) for j in (1..n))
flatten([[A023672(n,k) for k in (1..binomial(n+2,2))] for n in (1..10)]) # G. C. Greubel, Jul 18 2022

Formula

a(n) = Sum_{j=1..n} A000040(j) * A023533(n-j+1).

a(n) = Sum_{j=1..n} A000040(k + binomial(n+3, 3) - binomial(j+2, 3)), for 1 <= k <= binomial(n+2, 2), n >= 1 (as an irregular triangle). - G. C. Greubel, Jul 18 2022