A100289 Numbers k such that (1!)^2 + (2!)^2 + (3!)^2 + ... + (k!)^2 is prime.

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

Offset

1,1

Comments

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.

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]] &]
Module[{nn=350, tt}, tt=Accumulate[(Range[nn]!)^2]; Position[tt, _?PrimeQ]]//Flatten (* The program generates the first 16 terms of the sequence. *) (* Harvey P. Dale, Oct 12 2023 *)

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,more

Author

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

Extensions

a(18) from T. D. Noe, Feb 15 2006

a(19) from Serge Batalov, Jul 29 2017

Status

approved