A290014 -id:A290014 - cp's OEIS frontend

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

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

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.

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

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.

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

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 *)

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

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

a(19) from Serge Batalov, Jul 29 2017

A289947 Values of n for which Sum_{k=1..n} k!^6 is prime.

Original entry on oeis.org

5, 34, 102

Offset: 1

Eric W. Weisstein, Jul 16 2017

A289946(n) is divisible by 1091 for n >= 1090, and checking the terms below that gives A289946(a(3)) = A289946(102) as the final prime in the sequence.

			A289946(5) = 2986175149697 is prime.

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

Cf. A289946 (Sum_{k=1..n} k!^6).

Cf. A100289 (k!^2), A290014 (k!^10).

isok(n) = isprime(sum(k=1, n, k!^6)); \\ Michel Marcus, Jul 17 2017

A290250 Smallest (prime) number a(n) > 2 such that Sum_{k=1..a(n)} k!^(2*n) is divisible by a(n).

Original entry on oeis.org

1248829, 13, 1091, 13, 41, 37, 463, 13, 23, 13, 1667, 37, 23, 13, 41, 13, 139

Offset: 1

Eric W. Weisstein, Jul 24 2017

If a(i) exists, then the number of primes in the sequence {Sum_{k=1..n} k!^(2*i)}_n is finite.  This follows since all subsequent terms in the sum involve adding (1*2*...*a(i)*...)^(2*i) to the previous term, both of which are divisible by a(i).
The terms from a(19) to a(36) are 46147, 13, 587, 13, 107, 23, 41, 13, 163, 13, 43, 37, 23, 13, 397, 13, 23, 433, and the terms from a(38) to a(50) are 13, 419, 13, 9199, 23, 2129, 13, 41, 13, 2358661, 37, 409, 13. If they exist, a(18) > 25*10^6 and a(37) > 14*10^6. - Giovanni Resta, Jul 27 2017
a(37) = 17424871; a(18) > 5*10^7 - Mark Rodenkirch, Sep 04 2017

			sum(k=1..1248829, k!^2) = 14+ million-digit number which is divisible by 1248829
sum(k=1..13, k!^4) = 1503614384819523432725006336630745933089, which is divisible by 13
sum(k=1..1091, k!^6) = 17055-digit number which is divisible by 1091

Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A100289 (n such that Sum_{k=1..n} k!^2 is prime), A289945 (k!^4), A289946 (k!^6), A290014 (k!^10).

Table[Module[{sum = 1, fac = 1, k = 2}, While[! Divisible[sum += (fac *= k)^(2 n), k], k++]; k], {n, 17}]

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

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A289947 Values of n for which Sum_{k=1..n} k!^6 is prime.

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A290250 Smallest (prime) number a(n) > 2 such that Sum_{k=1..a(n)} k!^(2*n) is divisible by a(n).

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