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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 (list; graph; refs; listen; history; text; internal format)
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,p-1)-1, p divides S(2,n)-1 for all n >= p-1. Hence there are no primes for n >= p-1.

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

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)