A357813 a(n) is the least number k such that the sum of n^2 consecutive primes starting at prime(k) is a square.

3, 1, 78, 333, 84, 499, 36, 1874, 1102, 18, 183, 2706, 23, 104, 739, 1055, 8435, 633, 42130, 13800, 942, 55693, 7449, 13270, 41410, 4317, 17167, 61999, 17117, 9161, 46704, 12447, 2679, 2971, 3946, 103089, 6359, 19601, 7240, 422, 690, 20851, 963, 36597, 3559, 111687, 12926, 4071, 30622, 6355

Offset: 2

Author

Jean-Marc Rebert, Nov 12 2022

Keywords

• nonn

Examples

			Define sp(k,n) to be the sum of n^2 consecutive primes starting at prime(k).
a(2) = 3 because sp(k,2) at k=3 is 5 + 7 + 11 + 13 = 36 = 6^2, a square, and no smaller k has this property.
a(3) = 1 because sp(k,3) at k=1 is 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 = 10^2, a square, and no smaller k has this property.
a(4) = 78 because sp(k,4) at k=78 is 397 + 401 + ... + 487 = 7056 = 84^2, a square, and no smaller k has this property.
		

Crossrefs

Cf. A034963, A127336.

Cf. A358156. Subsequence of A230327.

Programs

PARI

\\ sum of n^2 consecutive primes starting at prime(k).
sp(k,n)=my(u=primes([prime(k),prime(k+n*n-1)]));return(vecsum(u))
\\ Least number k such that sp(k,n) is a square.
a(n)=my(k=1);while(!issquare(sp(k,n)),k++);k

PARI

a(n) = { my(pr = primes(n^2), s = vecsum(pr), startprime = nextprime(pr[#pr] + 1), res = 1); pr = List(pr); forprime(p = startprime, oo, if(issquare(s), return(res); ); res++; s += (p - pr[1]); listput(pr, p); listpop(pr, 1); ) } \\ David A. Corneth, Nov 13 2022

Formula

a(n) = A230327(n^2).