A287027 Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.

100, 961, 36, 14017536, 484, 49, 36, 134689, 354025, 80089, 443556, 121, 47524, 7744, 100, 700569, 344956329, 48841, 5329, 144, 324, 39601, 22801, 8649, 239438955625, 12250000, 197136, 222784, 147456, 319225, 316969, 24649, 576, 2975625, 7396, 21316, 70036245333532859364

Offset: 1

Author

Zak Seidov, May 18 2017

Keywords

• nonn

Comments

Squares that are the sum of 4 consecutive primes: 36, 324, 576, 1764, 2304, 4900, 20736, 63504, 66564, 128164, 142884, 150544, 156816, 183184, 236196, 256036, 260100, 311364, 369664, 414736.

Squares that are the sum of 5 consecutive primes: 961, 1089, 1681, 17689, 18769, 21025, 23409, 45369, 76729, 80089, 97969, 124609, 218089, 235225, 290521, 421201, 434281.

Squares that are the sum of 6 consecutive primes: 3600, 24336, 25600, 47524, 66564, 98596, 129600, 138384, 228484, 236196, 331776, 379456, 404496, 490000, 559504.

Squares that are the sum of 7 consecutive primes: 169, 625, 2209, 10201, 25921, 235225, 342225, 361201, 380689, 383161, 426409, 508369, 531441, 537289, 543169, 564001, 603729.

Note that A007504(m) - A007504(n) ~ m^2 log(m)/2 as m -> infinity. Heuristically this has probability ~ 1/(m sqrt(2 log(m))) of being a square. Since the sum of these probabilities diverges, on the basis of the second Borel-Cantelli lemma we should expect a(n) to exist. Of course, this is not a proof. Moreover, since the sum diverges very slowly, we might expect some very large values of a(n). - Robert Israel, May 18 2017

Examples

			Sum of set {2,3,5,7,11,13,17,19,23} is 100 = 10^2, sum of set {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83} is 961=31^2, sum of set {5,7,11,13}=36=6^2.
		

Crossrefs

Cf. A062703 (squares that are the sum of 2 consecutive primes), A080665 (squares that are the sum of 3 consecutive primes), A034707 (numbers that are sums of consecutive primes).

Cf. A007504.

Programs

Maple

f:= proc(n) local p, s;
  p:= ithprime(n); s:= p;
while not issqr(s) do p:= nextprime(p); s:= s+p od:
  s
end proc:
map(f, [$1..36]); # Robert Israel, May 18 2017

Mathematica

Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* Michael De Vlieger, May 20 2017 *)

PARI

a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ Felix Fröhlich, May 25 2017

Extensions

Missing a(25) and a(37) from Giovanni Resta, May 18 2017