1,1

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

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.

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

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

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.

Sequence in context: A233628 A030024 A232657 * A326638 A284515 A068086

Adjacent sequences: A287024 A287025 A287026 * A287028 A287029 A287030

Zak Seidov, May 18 2017

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

approved