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A287027
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Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* Michael De Vlieger, May 20 2017 *)
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PROG
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(PARI) a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ Felix Fröhlich, May 25 2017
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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