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%I #38 Sep 20 2018 00:31:12
%S 586,6088,8174,11585,11707,270106,288818,375661,724909,732910,937423,
%T 1141509,1326970,1619934,1776809,1930140,2239367,2489647,3063687,
%U 3649371,3790381,3941615,4193988,4821615,4887146,5572173,6047246,6192322,8088524,9158347
%N Numbers m such that m^2 is the sum of the squares of two or more consecutive primes.
%C m^2 = Sum_{i=k..j} prime(i)^2 is a square, for some k,j, j > k.
%C The 30 numbers given above are the only m values for all possible summations where the resulting m^2 < 10^14 (m <10^7). This requires searching from k values up to ~482,000, but with decreasing j-k ranges for efficiency.
%C Values of k that yield results begin: 13, 37, 101, 183, 235, 588, 805, 891, 1066, ... but do not correspond fully to the order of the m values shown.
%C Number of sequential summands (i.e., j-k+1) vary widely, with the smallest being 28 and largest being 10360, for those m values listed above.
%C Also note j-k+1 mod 8 = {0,1,4}, as expected, since prime(i)^2 mod 24 = 1, for i > 2.
%H Giovanni Resta, <a href="/A270424/b270424.txt">Table of n, a(n) for n = 1..164</a> (terms < 4*10^9)
%H Giovanni Resta, <a href="/A270424/a270424.txt">Details of the sums, for a(n) < 4*10^9</a>
%e 586 is in the sequence because 586^2 = 343396 = Sum_{i=13..40} prime(i)^2.
%t lim = 20000^2; L={}; P=Prime[Range[2 + PrimePi@ Sqrt[lim/2]]]^2; i = 1; While[ P[[i]] + P[[i+1]] <= lim, s = P[[i]]; j = i+1; While[(s += P[[j++]]) <= lim,If[IntegerQ@ Sqrt@ s, AppendTo[L, Sqrt@ s]]]; i++]; Union@L (* _Giovanni Resta_, Apr 13 2016 *)
%t result = {}; k = 3; While[k <= 481167, resultk = {}; sump = 0;
%t count = 0; i = k; While[sump < 10^14, sump += Prime[i]^2;
%t If[Mod[i - k + 1, 8] == 1 || Mod[i - k + 1, 8] == 0 ||
%t Mod[i - k + 1, 8] == 4, If[i != k && IntegerQ[Sqrt[sump]], count++;
%t AppendTo[resultk, {k, i - k + 1, sump}]]]; i++];
%t If[count > 0, AppendTo[result, resultk]]; k++]; result (* Only for k>2, so as to use index values to reduce repeated checking Sqrt - _Richard R. Forberg_, Apr 14 2016 *)
%Y Cf. A001248, A061890.
%K nonn
%O 1,1
%A _Richard R. Forberg_, Mar 30 2016
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