

A100289


Numbers n such that (1!)^2 + (2!)^2 + (3!)^2 +...+ (n!)^2 is prime.


8



2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841
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OFFSET

1,1


COMMENTS

All n <= 310 yield provable primes.
Write the sum as S(2,n)1, where S(k,n) = sum_{i=0..n} (i!)^k. Let p=1248829. Because p divides S(2,p1)1, p divides S(2,n)1 for all n >= p1. Hence there are no primes for n >= p1.


LINKS

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


MAPLE

L:= [seq((i!)^2, i=1..1000)]:
S:= ListTools:PartialSums(L):
select(t > isprime(S[t]), [$1..1000]); # Robert Israel, Jul 17 2017


MATHEMATICA

Select[Range[200], PrimeQ[Total[Range[#]!^2]] &]


PROG

(PARI) is(n)=ispseudoprime(sum(k=1, n, k!^2)) \\ Charles R Greathouse IV, Apr 14 2015


CROSSREFS

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


KEYWORD

nonn,fini


AUTHOR

T. D. Noe, Nov 11 2004 and Dec 11 2004


EXTENSIONS

a(18) from T. D. Noe, Feb 15 2006
a(19) = 32841 from Serge Batalov, Jul 29 2017


STATUS

approved



