A270424 Numbers m such that m^2 is the sum of the squares of two or more consecutive primes.

586, 6088, 8174, 11585, 11707, 270106, 288818, 375661, 724909, 732910, 937423, 1141509, 1326970, 1619934, 1776809, 1930140, 2239367, 2489647, 3063687, 3649371, 3790381, 3941615, 4193988, 4821615, 4887146, 5572173, 6047246, 6192322, 8088524, 9158347

Offset: 1

Author

Richard R. Forberg, Mar 30 2016

Keywords

• nonn

Comments

m^2 = Sum_{i=k..j} prime(i)^2 is a square, for some k,j, j > k.

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.

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.

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.

Also note j-k+1 mod 8 = {0,1,4}, as expected, since prime(i)^2 mod 24 = 1, for i > 2.

Examples

			586 is in the sequence because 586^2 = 343396 = Sum_{i=13..40} prime(i)^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..164 (terms < 4*10^9)
• Giovanni Resta, Details of the sums, for a(n) < 4*10^9

Crossrefs

Cf. A001248, A061890.

Programs

Mathematica

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 *)
result = {}; k = 3; While[k <= 481167, resultk = {}; sump = 0;
count = 0; i = k; While[sump < 10^14, sump += Prime[i]^2;
  If[Mod[i - k + 1, 8] == 1 || Mod[i - k + 1, 8] == 0 ||
    Mod[i - k + 1, 8] == 4, If[i != k && IntegerQ[Sqrt[sump]], count++;
    AppendTo[resultk, {k, i - k + 1, sump}]]]; i++];
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 *)