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

Author

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

Keywords

• nonn
• fini
• more

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

• Eric Weisstein's World of Mathematics, Factorial Sums
• Eric Weisstein's World of Mathematics, Integer Sequence Primes

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.

Programs

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 *)

PARI

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

Extensions

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

a(19) from Serge Batalov, Jul 29 2017