A230327 -id:A230327 - cp's OEIS frontend

A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

9, 23, 4, 1862, 14, 3, 2, 211, 331, 163, 366, 3, 124, 48, 2, 449, 8403, 121, 35, 2, 4, 105, 77, 43, 190769, 1726, 234, 248, 200, 295, 293, 73, 4, 873, 32, 64, 2456139382, 8, 4519, 14, 123, 5, 9395, 296, 26, 5, 3479, 810, 9, 7091, 1669, 157, 1189, 12559, 269, 4930, 21, 376, 3

Offset: 1

Todor Szimeonov, Nov 01 2022

nonn

a(60) > 10^10 and a(68) > 10^13. - Martin Ehrenstein, Nov 09 2022

			For n=7, prime(7) = 17 and starting there 2 primes 17 + 19 = 36 which is square, so that a(7)=2.

Todor Szimeonov, Square of prime numbers

Cf. A000040, A000290, A105720, A230327 (exchanges the roles of n, k), A287027 (squares reached).

Indices of terms: A064397 (2's), A076305 (3's), A072849 (4's), A166255 (70's), A166261 (120's).

f:= proc(n) local p,s,k;
  p:= ithprime(n); s:= p;
  for k from 2 do
    p:= nextprime(p);
    s:= s+p;
    if issqr(s) then return k fi
  od
end proc:
map(f, [$1..36]); # Robert Israel, Nov 08 2022

a[n_] := Module[{p = s = Prime[n], k = 1}, While[! IntegerQ[Sqrt[s]], p = NextPrime[p]; s += p; k++]; k]; Array[a, 36] (* Amiram Eldar, Nov 08 2022 *)

a(25)-a(36) from Robert Israel, Nov 08 2022

a(37)-a(59) from Martin Ehrenstein, Nov 09 2022

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

Original entry on oeis.org

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

Jean-Marc Rebert, Nov 12 2022

nonn

			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.

Cf. A034963, A127336.

Cf. A358156. Subsequence of A230327.

\\ 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

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

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

cp's OEIS Frontend

A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

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