A289947 -id:A289947 - 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

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

Original entry on oeis.org

1, 65, 46721, 191149697, 2986175149697, 139317055679149697, 16390300280131775149697, 4296598745804900241599149697, 2283384320190476620685217983149697, 2283382306976051006261597069217983149697

Offset: 1

Eric W. Weisstein, Jul 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..104
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. [This thesis cites this sequence entry, but it's just a typo: the intended sequence entry is A289546.]
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A007489 (k!), A104344 (k!^2), A289945 (k!^4).

Cf. A289947 (indices giving primes).

Table[Sum[k!^6, {k, n}], {n, 10}]
Accumulate[(Range[10]!)^6] (* Harvey P. Dale, May 14 2023 *)

a(n) = sum(k=1, n, k!^6); \\ Michel Marcus, Jul 16 2017

A290014 Values of n for which Sum_{k=1..n} k!^10 is prime.

Original entry on oeis.org

3, 4, 5, 16, 25

Offset: 1

Eric W. Weisstein, Jul 17 2017

Sum_{k=1..n} k!^10 is divisible by 41 for n >= 40, and checking the terms below that gives Sum_{k=1..a(5)} k!^10 with a(5) = 25 as the final prime in the sequence.

			Sum_{k=1..3} k!^10 = 60467201 is prime.
Sum_{k=1..4} k!^10 = 63403441432577 is prime.
Sum_{k=1..5} k!^10 = 619173705643441432577 is prime.
...

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

Cf. A100289 (k!^2), A289947 (k!^6).

cp's OEIS Frontend

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

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

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A290014 Values of n for which Sum_{k=1..n} k!^10 is prime.

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