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).

2, 3, 16, 21, 23, 39, 49, 381, 396, 24963, 39762, 40101, 40276, 4431583, 21553054, 36244531, 2183957515, 2183971285, 2183971945, 3636636400, 3636636411, 6063744535, 16846463635, 28070695902, 215867952637, 359222008925, 597739400517, 597739400913, 597739426757

Offset: 1

Author

Ctibor O. Zizka, Apr 24 2025

Keywords

• nonn

Examples

			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.
		

References

• Kenichiro Kashihara, Problem 10 in Comments and topics on Smarandache notions and problems, Erhus University Press, USA, 1996.

Links

• Xiaoxia Yan, On the Smarandache prime part, Scientia Magna, Vol. 3, No. 3 (2007), pp. 74-77.

Crossrefs

Cf. A007917, A007918.

Programs

Mathematica

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 *)

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

Extensions

a(10)-a(23) from Amiram Eldar, Apr 24 2025

a(24)-a(29) from Jinyuan Wang, Apr 28 2025