A289946 -id:A289946 - cp's OEIS frontend

A104344 a(n) = Sum_{k=1..n} k!^2.

1, 5, 41, 617, 15017, 533417, 25935017, 1651637417, 133333531817, 13301522971817, 1606652445211817, 231049185247771817, 39006837228880411817, 7639061293780877851817, 1717651314017980301851817, 439480788011413032845851817, 126953027293558583218061851817

Offset: 1

Eric W. Weisstein, Mar 02 2005

Seiichi Manyama, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A001044, A061062, A100289.

Sum_{k=1..n} (k!)^m: A007489 (m=1), this sequence (m=2), A138564 (m=3), A289945 (m=4), A316777 (m=5), A289946 (m=6).

Table[Sum[(k!)^2,{k,n}],{n,15}] (* Harvey P. Dale, Jul 21 2011 *)
Accumulate[(Range[20]!)^2] (* Much more efficient than the above program. *) (* Harvey P. Dale, Aug 15 2022 *)

a(n) = sum(k=1, n, k!^2); \\ Michel Marcus, Jul 16 2017

a(n) = A061062(n) - 1. - Michel Marcus, Feb 28 2014

More terms from Vladimir Joseph Stephan Orlovsky, Sep 24 2009

A289945 a(n) = Sum_{k=1..n} k!^4.

Original entry on oeis.org

1, 17, 1313, 333089, 207693089, 268946253089, 645510228813089, 2643553803594573089, 17342764866576345933089, 173418555892594089945933089, 2538940579958951120707545933089, 52646414799433780559063261145933089

Offset: 1

Eric W. Weisstein, Jul 16 2017

nonn

The only prime in this sequence is a(2) = 17 since a(n) is divisible by 13 for n >= 12 and there are no other primes with n < 12.

Seiichi Manyama, Table of n, a(n) for n = 1..144
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A007489 (k!), A104344 (k!^2), A289946 (k!^6).

Table[Sum[k!^4, {k, n}], {n, 12}]
Accumulate[(Range[15]!)^4] (* Harvey P. Dale, Jul 12 2019 *)

a(n) = sum(k=1, n, k!^4); \\ Michel Marcus, Jul 16 2017

A138564 a(1) = 1; a(n) = a(n-1) + (n!)^3.

Original entry on oeis.org

1, 9, 225, 14049, 1742049, 374990049, 128399054049, 65676719822049, 47850402559694049, 47832576242431694049, 63649302669112063694049, 109966989623147836159694049, 241567605673714904675071694049

Offset: 1

Jonathan Vos Post, May 18 2008

By sum of cubes factorization, every a(n) > 1 is a multiple of 9, hence none of these are prime, unlike the case of sum of squares of factorials (i.e. (1!)^2 + (2!)^2+ (3!)^2+ (4!)^2 = 617 is prime; 41117342095090841723228045851817 = (1!)^2 + (2!)^2 + (3!)^2 + (4!)^2 + (5!)^2 + (6!)^2 + (7!)^2 + (8!)^2 + (9!)^2 + (10!)^2 + (11!)^2 + (12!)^2 + (13!)^2 + (14!)^2 + (15!)^2 + (16!)^2 + (17!)^2 + (18!)^2 is prime).

			a(18) = (1!)^3 + (2!)^3 + (3!)^3 + (4!)^3 + (5!)^3 + (6!)^3 + (7!)^3 + (8!)^3 + (9!)^3 + (10!)^3 + (11!)^3 + (12!)^3 + (13!)^3 + (14!)^3 + (15!)^3 + (16!)^3 + (17!)^3 + (18!)^3 = 262480797594664584673157017306412926841599694049.

Seiichi Manyama, Table of n, a(n) for n = 1..181

Cf. A000142, A000578, A100288.

Sum_{k=1..n} (k!)^m: A007489 (m=1), A104344 (m=2), this sequence (m=3), A289945 (m=4), A289946 (m=6).

nxt[{n_,a_}]:={n+1,a+((n+1)!)^3}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Mar 08 2015 *)

a(n) = Sum_{k=1..n} (k!)^3 = Sum_{k=1..n} A000578(A000142(n)).

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

A316777 a(n) = Sum_{k=1..n} (k!)^5.

Original entry on oeis.org

1, 33, 7809, 7970433, 24891170433, 193516654370433, 3252209580756770433, 106565314598088199970433, 6292489787293574101716770433, 629244614687684894925781716770433, 101340090272908294280184341461716770433, 25216602106682627624613336066827221716770433

Offset: 1

Seiichi Manyama, Aug 03 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, Factorial Sums

Sum_{k=1..n} (k!)^m: A007489 (m=1), A104344 (m=2), A138564 (m=3), A289945 (m=4), this sequence (m=5), A289946 (m=6).

Accumulate[(Range[15]!)^5] (* Harvey P. Dale, Jan 24 2021 *)

PARI
```
{a(n) = sum(k=1, n, k!^5)}
```

A339311 a(n) = Sum_{k=1..n} (k!)^n.

Original entry on oeis.org

1, 5, 225, 333089, 24891170433, 139317055679149697, 82606511560391889386751489, 6984964663477899063759529684476994049, 109110688697205077023425816926479750793809817618433, 395940866162019279867473563258123776010342586429441374123441432577

Offset: 1

Seiichi Manyama, Nov 29 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..30

Sum_{k=1..n} (k!)^m: A007489 (m=1), A104344 (m=2), A138564 (m=3), A289945 (m=4), A316777 (m=5), A289946 (m=6).

Cf. A000142, A112999.

a[n_] := Sum[(k!)^n, {k, 1, n}]; Array[a, 10] (* Amiram Eldar, Apr 28 2021 *)

PARI
```
{a(n) = sum(k=1, n, k!^n)}
```

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}]

cp's OEIS Frontend

A104344 a(n) = Sum_{k=1..n} k!^2.

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A289945 a(n) = Sum_{k=1..n} k!^4.

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A138564 a(1) = 1; a(n) = a(n-1) + (n!)^3.

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

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A316777 a(n) = Sum_{k=1..n} (k!)^5.

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PARI

A339311 a(n) = Sum_{k=1..n} (k!)^n.

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Mathematica

PARI

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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