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%I #57 Sep 08 2022 08:44:38
%S 1,2,4,6,12,14,60,64,96,100,102,108,114,122,124,130,132,146,152,158,
%T 162,178,192,198,204,206,208,214,216,296,308,326,328,330,332,334,342,
%U 350,356,358,426,446,458,460,464,480,484,488,512,530,536,548,568,620,630,676,680
%N Numbers k such that the sum of the first k primes is prime.
%H David W. Wilson, <a href="/A013916/b013916.txt">Table of n, a(n) for n = 1..10000</a>
%H Romeo Meštrović, <a href="https://arxiv.org/abs/1804.04198">Curious conjectures on the distribution of primes among the sums of the first 2n primes</a>, arXiv:1804.04198 [math.NT], 2018.
%F a(n) = A000720(A013917(n)).
%e 6 is a term because the sum of the first six primes 2 + 3 + 5 + 7 + 11 + 13 = 41 is prime.
%p p:=proc(n) if isprime(sum(ithprime(k),k=1..n))=true then n else fi end: seq(p(n),n=1..690); # _Emeric Deutsch_
%t s = 0; Do[s = s + Prime[n]; If[PrimeQ[s], Print[n]], {n, 1, 1000}]
%t Flatten[Position[Accumulate[Prime[Range[2000]]], _?(PrimeQ[#] &)]] (* _Harvey P. Dale_, Dec 16 2010 *)
%t Flatten[Position[PrimeQ[Accumulate[Prime[Range[2000]]]],True]] (* _Fred Patrick Doty_, Aug 15 2017 *)
%o (PARI) isA013916(n) = isprime(sum(i=1,n,prime(i))) \\ _Michael B. Porter_, Jan 29 2010
%o (Magma) [n:n in [1..700] | IsPrime(&+PrimesUpTo(NthPrime(n))) ]; // _Marius A. Burtea_, Jan 04 2019
%o (MATLAB) p=primes(10000); m=1;
%o for u=1:700 ; suma=sum(p(1:u));
%o if isprime(suma)==1 ; sol(m)=u; m=m+1; end
%o end
%o sol; % _Marius A. Burtea_, Jan 04 2019
%o (GAP) P:=Filtered([1..5300],IsPrime);;
%o a:=Filtered([1..Length(P)],n->IsPrime(Sum([1..n],k->P[k])));; Print(a); # _Muniru A Asiru_, Jan 04 2019
%o (Python)
%o from sympy import isprime, prime
%o def aupto(lim):
%o s = 0
%o for k in range(1, lim+1):
%o s += prime(k)
%o if isprime(s): print(k, end=", ")
%o aupto(680) # _Michael S. Branicky_, Feb 28 2021
%Y Cf. A007504, A013917, A013918.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_, Renaud Lifchitz (100637.64(AT)CompuServe.COM)
%E More terms from _David W. Wilson_
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