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Define sp(k,n) to be the sum of n^3 consecutive primes starting at prime(k). Then a(n) is the least number k such that sp(k,n) is a cube, or -1 if no such number exists.
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%I #28 Dec 21 2022 22:09:40

%S 2704,74,734,19189898,26509715,69713,4521289,2173287,2785343,

%T 228207824,570319147,5229039,57210987

%N Define sp(k,n) to be the sum of n^3 consecutive primes starting at prime(k). Then a(n) is the least number k such that sp(k,n) is a cube, or -1 if no such number exists.

%e a(2) = 2704 because sp(k,2) at k = 2794 is prime(2704) + prime(2705) + ... + prime(2704 + 2^3 - 1) = 24359 + 24371 + ... + 24419 = 195112 = 58^3, a cube, and no smaller k has this property.

%e a(3) = 74 because sp(k,3) at k = 74 is prime(74) + prime(75) + ... + prime(74 + 3^3 - 1) = 373 + 379 + ... + 541 = 12167 = 23^3, a cube, and no smaller k has this property.

%o (PARI)

%o a(n)=my(k=1,vp=primes(n^3),s=vecsum(vp));while(!ispower(s,3),p=nextprime(vp[#vp]+1);s+=(p-vp[1]);vp=concat(vp,p);vp=vp[^1];k++);k

%Y Cf. A127335.

%Y Cf. A358156, A357813.

%K nonn,more

%O 2,1

%A _Jean-Marc Rebert_, Nov 15 2022