This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383358 #33 May 10 2025 08:59:47
%S A383358 2,3,16,21,23,39,49,381,396,24963,39762,40101,40276,4431583,21553054,
%T A383358 36244531,2183957515,2183971285,2183971945,3636636400,3636636411,
%U A383358 6063744535,16846463635,28070695902,215867952637,359222008925,597739400517,597739400913,597739426757
%N A383358 Numbers k >= 2 such that (S(k) - I(k)) / (k - 1) is an integer, where S(k) = Sum_{i=2..k} A007918(i) and I(k) = Sum_{i=2..k} A007917(i).
%D A383358 Kenichiro Kashihara, Problem 10 in Comments and topics on Smarandache notions and problems, Erhus University Press, USA, 1996.
%H A383358 Xiaoxia Yan, <a href="https://mail.vixra.org/pdf/1403.0432v1.pdf">On the Smarandache prime part</a>, Scientia Magna, Vol. 3, No. 3 (2007), pp. 74-77.
%e A383358 k = 16: ((Sum_{i=2..k} A007918(i)) - (Sum_{i=2..k} A007917(i))) / (k - 1) = (150 - 120) / 15 = 2, thus k = 16 is a term.
%t A383358 With[{m = 10^7}, (Position[Accumulate[Table[If[PrimeQ[n], 0, NextPrime[n] - NextPrime[n, -1]], {n, 2, m+1}]]/Range[m], _?IntegerQ] // Flatten) + 1] (* _Amiram Eldar_, Apr 24 2025 *)
%o A383358 (PARI) list(lim) = {my(k = 0, c = 0, p = 2, g); forprime(q = 3, lim, g = q-p; k++; if(!(c % k), print1(k+1, ", ")); for(i = 1, g-1, c += g; k++; if(!(c % k), print1(k+1, ", "))); p = q);} \\ _Amiram Eldar_, Apr 24 2025
%Y A383358 Cf. A007917, A007918.
%K A383358 nonn
%O A383358 1,1
%A A383358 _Ctibor O. Zizka_, Apr 24 2025
%E A383358 a(10)-a(23) from _Amiram Eldar_, Apr 24 2025
%E A383358 a(24)-a(29) from _Jinyuan Wang_, Apr 28 2025