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A100289
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Numbers k such that (1!)^2 + (2!)^2 + (3!)^2 + ... + (k!)^2 is prime.
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8
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2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841
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OFFSET
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1,1
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COMMENTS
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All k <= 310 yield provable primes.
Write the sum as S(2,k)-1, where S(m,k) = Sum_{i=0..k} (i!)^m. Let p=1248829. Because p divides S(2,p-1)-1, p divides S(2,k)-1 for all k >= p-1. Hence there are no primes for k >= p-1.
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LINKS
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Table of n, a(n) for n=1..19.
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Integer Sequence Primes
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MAPLE
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L:= [seq((i!)^2, i=1..1000)]:
S:= ListTools:-PartialSums(L):
select(t -> isprime(S[t]), [$1..1000]); # Robert Israel, Jul 17 2017
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MATHEMATICA
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Select[Range[200], PrimeQ[Total[Range[#]!^2]] &]
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PROG
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(PARI) is(n)=ispseudoprime(sum(k=1, n, k!^2)) \\ Charles R Greathouse IV, Apr 14 2015
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CROSSREFS
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Cf. A100288 (primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2).
Cf. A061062 ((0!)^2 + (1!)^2 + (2!)^2 + (3!)^2 +...+ (n!)^2).
Cf. A289947 (k!^6), A290014 (k!^10).
Cf. also A104344.
Sequence in context: A034296 A075745 A214036 * A255130 A054021 A066191
Adjacent sequences: A100286 A100287 A100288 * A100290 A100291 A100292
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KEYWORD
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nonn,fini,more
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AUTHOR
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T. D. Noe, Nov 11 2004 and Dec 11 2004
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EXTENSIONS
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a(18) from T. D. Noe, Feb 15 2006
a(19) from Serge Batalov, Jul 29 2017
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STATUS
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approved
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