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%I #30 Aug 10 2023 03:32:48
%S 1,441,970225,1464100,194379364,1303400915339554201
%N Powers equal to (sum of first k primes) plus 1, for some k >= 0.
%C I have checked for powers out to the 250000th prime and the last element found is at the 6420th prime, 64067. It is interesting to note that the only powers so far are squares. Finding a higher power would be interesting.
%e 1 is a term (corresponding to k=0), since it is the empty sum plus 1. - _N. J. A. Sloane_, Dec 02 2015
%e 441 is a term since sum(primes<=59) = 440 and 441 = 21^2.
%p with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: s := proc(n) option remember; local p; if n=1 then [1,2] else [n,s(n-1)[2]+ithprime(n)] fi end; t := proc(n) option remember; [n,s(n)[2]+1] fi end; PW:=[]; for z to 1 do for j from 1 to 250000 do if egcd(t(j)[2])>1 then PW:=[op(PW),t(j)] fi od od; PW;
%o (PARI) lista(nn) = { print1(1, ", "); s = 1; for(k=1, nn, s += prime(k); if(ispower(s) || s==1, print1(s, ", ")););} \\ _Altug Alkan_, Nov 29 2015
%Y Cf. A007504, A110979, A110997, A264858.
%Y Intersection of A001597 and A014284.
%K nonn,more
%O 1,2
%A _Walter Kehowski_, Sep 30 2005
%E New term 1 prepended by _Altug Alkan_, Nov 29 2015
%E a(6) from _Jinyuan Wang_, Aug 09 2023
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