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%I #35 Dec 30 2018 12:45:36
%S 8,21,27,37,59,65,66,82,86,99,105,111,126,143,147,155,156,165,177,181,
%T 187,194,195,200,230,231,242,262,284,374,430,449,460,477,502,512,539,
%U 540,541,622,634,657,707,731,735,739,745,766,767,781,784,785,791,801
%N Numbers k such that if j is the sum of the first prime(k) primes then the sum of the first j primes is prime.
%C k is a term if A007504(A007504(prime(k)) is prime. Terms can be even or odd since A007504(A007504(prime(k)) is odd for any k.
%H Ray Chandler, <a href="/A321439/b321439.txt">Table of n, a(n) for n = 1..2500</a>
%e 8 is a term because prime(8) = 19, A007504(19) = 568, and A007504(568) = 1086557, which is prime.
%e 2 is not a term since prime(2) = 3, A007504(3) = 10 and A007504(10) = 129, which is not prime.
%p N:=100:
%p for n from 1 to N do
%p X:=add(ithprime(k),k=1..ithprime(n));
%p Y:=add(ithprime(r),r=1..X);
%p if isprime(Y)then print(n);
%p end if:
%p end do:
%t primeSum[n_] := Sum[Prime[i], {i, n}]; Select[Range[200], PrimeQ[ primeSum[primeSum[Prime[#]]]] &] (* _Amiram Eldar_, Nov 09 2018 *)
%o (Perl)
%o use ntheory qw(:all);
%o for (my ($i, $k) = (1, 1); ; ++$k) {
%o if (is_prime sum_primes nth_prime sum_primes nth_prime nth_prime $k) {
%o print "a($i) = $k\n"; ++$i;
%o }
%o } # _Daniel Suteu_, Nov 11 2018
%o (PARI)
%o sumprimes(n)={my(p=0, s=0); for(i=1, n, p=nextprime(1+p); s+=p); s}
%o ok(k)={isprime(sumprimes(sumprimes(prime(k))))}
%o for(n=1, 100, if(ok(n),print1(n, ", "))) \\ _Andrew Howroyd_, Nov 11 2018
%Y Cf. A007504, A013916, A321342, A321343.
%K nonn
%O 1,1
%A _David James Sycamore_, Nov 09 2018
%E a(30)-a(54) from _Daniel Suteu_, Nov 11 2018
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