A358430 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.
2704, 74, 734, 19189898, 26509715, 69713, 4521289, 2173287, 2785343, 228207824, 570319147, 5229039, 57210987
Offset: 2
Examples
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. 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.
Programs
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PARI
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