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A023672 Convolution of A023533 and primes.

%I A023672 #7 Jul 19 2022 08:12:18
%S A023672 2,3,5,9,14,18,24,30,36,48,53,65,77,85,97,111,121,131,149,163,174,192,
%T A023672 204,220,242,260,272,294,310,320,350,364,382,410,436,453,469,495,513,
%U A023672 543,569,587,615,647,661,687,715,739,759,799,827,855,869
%N A023672 Convolution of A023533 and primes.
%H A023672 G. C. Greubel, <a href="/A023672/b023672.txt">Table of n, a(n) for n = 1..5000</a>
%F A023672 a(n) = Sum_{j=1..n} A000040(j) * A023533(n-j+1).
%F A023672 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 A023672[n_, m_]:= A023672[n, m]= Sum[Prime[(m +Binomial[n+2,3] -Binomial[j+2, 3])], {j, n}];
%t A023672 Table[A023672[n, m], {n,10}, {m,Binomial[n+2,2]}]//Flatten (* _G. C. Greubel_, Jul 18 2022 *)
%o A023672 (Magma)
%o A023672 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023672 [(&+[NthPrime(k)*A023533(n+1-k): k in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 18 2022
%o A023672 (SageMath)
%o A023672 def A023672(n,k): return sum(nth_prime(k +binomial(n+2,3) -binomial(j+2,3)) for j in (1..n))
%o A023672 flatten([[A023672(n,k) for k in (1..binomial(n+2,2))] for n in (1..10)]) # _G. C. Greubel_, Jul 18 2022
%Y A023672 Cf. A000040, A023533.
%K A023672 nonn
%O A023672 1,1
%A A023672 _Clark Kimberling_