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

%I A287027 #30 May 25 2017 14:50:13
%S A287027 100,961,36,14017536,484,49,36,134689,354025,80089,443556,121,47524,
%T A287027 7744,100,700569,344956329,48841,5329,144,324,39601,22801,8649,
%U A287027 239438955625,12250000,197136,222784,147456,319225,316969,24649,576,2975625,7396,21316,70036245333532859364
%N A287027 Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.
%C A287027 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.
%C A287027 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.
%C A287027 Squares that are the sum of 6 consecutive primes: 3600, 24336, 25600, 47524, 66564, 98596, 129600, 138384, 228484, 236196, 331776, 379456, 404496, 490000, 559504.
%C A287027 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.
%C A287027 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
%e A287027 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.
%p A287027 f:= proc(n) local p, s;
%p A287027   p:= ithprime(n); s:= p;
%p A287027 while not issqr(s) do p:= nextprime(p); s:= s+p od:
%p A287027   s
%p A287027 end proc:
%p A287027 map(f, [$1..36]); # _Robert Israel_, May 18 2017
%t A287027 Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* _Michael De Vlieger_, May 20 2017 *)
%o A287027 (PARI) a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ _Felix Fröhlich_, May 25 2017
%Y A287027 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).
%Y A287027 Cf. A007504.
%K A287027 nonn
%O A287027 1,1
%A A287027 _Zak Seidov_, May 18 2017
%E A287027 Missing a(25) and a(37) from _Giovanni Resta_, May 18 2017