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%I #22 Aug 25 2023 12:04:07
%S 3,3833,468872968241
%N Prime numbers p for which there exists an integer q > p such that the sum of all primes <= p equals the sum of all primes between p+1 and q.
%C Primes p such that 2*A034387(p) is a term of A034387. - _Max Alekseyev_, Aug 24 2023
%C No other terms below 10^13. - _Max Alekseyev_, Aug 25 2023
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_018.htm">Puzzle 18.- Some special sums of consecutive primes</a>, The Prime Puzzles & Problems Connection. See specifically the large solution by Giovanni Resta.
%e 2+3+5+...+3833 = 3847+...+5557 and therefore 3833 is in the sequence.
%t a[m_] := Module[{pLst, cumsum, p, q, k, target, idx}, pLst = Prime[Range[PrimePi[m]]]; cumsum = Accumulate[pLst]; pairs = {}; For[k = 1, k <= Length[pLst], k++, p = pLst[[k]]; target = 2*cumsum[[k]]; idx = FirstPosition[Drop[cumsum, k], target]; If[idx =!= Missing["NotFound"], q = pLst[[k + First[idx]]]; If[q > p, AppendTo[pairs, p];]]]; pairs]; a[10000] (* _Robert P. P. McKone_, Aug 25 2023 *)
%o (PARI) p=2;s=2;q=3;t=3;while(p<512345678900, while(s<=t,p=nextprime(p+1);s=s+p;t=t-p);if (s==t,print1(p,", "),);while(t<s,q=nextprime(q+1);t=t+q);if(s==t,print1(p,", "),))
%K more,nonn,bref
%O 1,1
%A _Randy L. Ekl_, Jan 10 2004
%E Better definition from Adam M. Kalman (mocha(AT)clarityconnect.com), Jun 16 2005
%E Edited by _Max Alekseyev_, Aug 24 2023
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