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A321342 Numbers k such that if j is the sum of the first k primes, then the sum of the first j primes is prime.

%I A321342 #48 Dec 30 2018 12:39:35
%S A321342 1,9,15,19,73,85,87,103,121,157,175,277,313,317,341,357,375,385,391,
%T A321342 421,443,447,523,525,539,571,607,611,645,701,779,783,791,799,823,831,
%U A321342 835,853,889,907,911,925,977,1051,1075,1081,1087,1095,1117,1125,1135,1157,1181,1187,1223,1257,1305,1325
%N A321342 Numbers k such that if j is the sum of the first k primes, then the sum of the first j primes is prime.
%C A321342 k is in the sequence if A007504(j) is prime, where j = A007504(k). A007504(j) must be odd to be prime, so j must be even and k must be odd. Therefore all terms are odd. The subsequence of primes is A321343.
%H A321342 Ray Chandler, <a href="/A321342/b321342.txt">Table of n, a(n) for n = 1..16000</a>
%H A321342 Daniel Suteu, <a href="/A321342/a321342.pl.txt">Perl program</a>
%e A321342 A007504(1) = 2 and A007504(2) = 5, a prime therefore 1 is a term.
%e A321342 A007504(3) = 10 and A007504(10) = 129, not prime, therefore 3 is not a term.
%e A321342 A007504(9) = 100 and A007504(100) = 24133, a prime so 9 is a term.
%p A321342 N:=2000:
%p A321342 for n from 1 to N by 2 do
%p A321342 X:=add(ithprime(r),r=1..n);
%p A321342 Y:=add(ithprime(k),k=1..X);
%p A321342 if isprime(Y) then print(n);
%p A321342 end if:
%p A321342 end do:
%t A321342 primeSum[n_] := Sum[Prime[i], {i, n}]; Select[Range[300], PrimeQ[primeSum[primeSum[#]]] &] (* _Amiram Eldar_, Nov 07 2018 *)
%o A321342 (PARI) sfp(n) = sum(k=1, n, prime(k)); \\ A007504
%o A321342 isok(n) = isprime(sfp(sfp(n))); \\ _Michel Marcus_, Nov 08 2018
%Y A321342 Cf. A007504, A013916, A321343.
%K A321342 nonn
%O A321342 1,2
%A A321342 _David James Sycamore_, Nov 06 2018