A023672 - OEIS

%I #7 Jul 19 2022 08:12:18

%S 2,3,5,9,14,18,24,30,36,48,53,65,77,85,97,111,121,131,149,163,174,192,

%T 204,220,242,260,272,294,310,320,350,364,382,410,436,453,469,495,513,

%U 543,569,587,615,647,661,687,715,739,759,799,827,855,869

%N Convolution of A023533 and primes.

%H G. C. Greubel, <a href="/A023672/b023672.txt">Table of n, a(n) for n = 1..5000</a>

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

%F 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

%t A023672[n_, m_]:= A023672[n, m]= Sum[Prime[(m +Binomial[n+2,3] -Binomial[j+2, 3])], {j, n}];

%t Table[A023672[n, m], {n,10}, {m,Binomial[n+2,2]}]//Flatten (* _G. C. Greubel_, Jul 18 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o [(&+[NthPrime(k)*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 18 2022

%o (SageMath)

%o def A023672(n,k): return sum(nth_prime(k +binomial(n+2,3) -binomial(j+2,3)) for j in (1..n))

%o flatten([[A023672(n,k) for k in (1..binomial(n+2,2))] for n in (1..10)]) # _G. C. Greubel_, Jul 18 2022

%Y Cf. A000040, A023533.

%K nonn

%O 1,1

%A _Clark Kimberling_