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a(n) is smallest prime not already in the sequence such that the sum of the first n terms is an n-th power.
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%I #12 Dec 16 2024 19:18:35

%S 2,7,503,113,7151,109873,162287,2251875110689,1423309560546093919,

%T 1831931738588396657,266306005917953213327,

%U 13573982207668041076860262513,4485809513902758532742979512207

%N a(n) is smallest prime not already in the sequence such that the sum of the first n terms is an n-th power.

%H Sean A. Irvine, <a href="/A073698/b073698.txt">Table of n, a(n) for n = 1..100</a>

%e Sum of two terms 2+7 = 9 = 3^2. Sum of three terms 2+7+503 = 512 = 8^3.

%t a[n_] := a[n]=Module[{sm, i}, sm=Sum[a[i], {i, 1, n-1}]; For[k=Ceiling[(sm+2)^(1/n)], !ProvablePrimeQ[k^n-sm]||MemberQ[a/@Range[n-1], k^n-sm], k++, Null]; k^n-sm] (* First do <<NumberTheory`PrimeQ` *)

%Y Cf. A093927, A093928, A093929, A093355.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Aug 12 2002

%E Edited by _Dean Hickerson_, Dec 02 2002