A100288 -id:A100288 - cp's OEIS frontend

A061062 Sum of squared factorials: (0!)^2 + (1!)^2 + (2!)^2 + (3!)^2 +...+ (n!)^2.

1, 2, 6, 42, 618, 15018, 533418, 25935018, 1651637418, 133333531818, 13301522971818, 1606652445211818, 231049185247771818, 39006837228880411818, 7639061293780877851818, 1717651314017980301851818

Offset: 0

Jason Earls, May 27 2001

nonn

There is a Kurepa-like conjecture (see A049782) for this sequence: for primes p>3, p does not divide a(p-1). However, the conjecture fails for p=20879. - T. D. Noe, Dec 08 2004

			a(2) = 0!*0! + 1!*1! + 2!*2! = 6.

Seiichi Manyama, Table of n, a(n) for n = 0..253 (terms 0..100 from Harry J. Smith)

Cf. A001044, A100288 (primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2), A104344 (if sum starts at k=1), A049782.

A061062:=n->sum((k!)^2, k=0..n): seq(A061062(n), n=0..15); # Zerinvary Lajos, Jan 22 2008

s=0; Table[s=s+(n!)^2, {n, 0, 20}]
Accumulate[(Range[0,20]!)^2] (* Harvey P. Dale, Apr 19 2015 *)

{ a=0; for (n=0, 100, write("b061062.txt", n, " ", a+=(n!)^2) ) } \\ Harry J. Smith, Jul 17 2009

a(n) = sum(k=0...n, (n-k)!^2 ). - Ross La Haye, Sep 21 2004

Recurrence: a(0) = 1, a(1) = 2, a(n) = (n^2+1)*a(n-1) - n^2*a(n-2). - Vladimir Reshetnikov, Oct 28 2015

More terms from T. D. Noe, Dec 08 2004

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

Original entry on oeis.org

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

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)).

A101746 Primes of the form ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6.

Original entry on oeis.org

7, 103, 2503, 88903, 4322503, 2473107965928318342544472044975303

Offset: 1

T. D. Noe, Dec 18 2004

Let S(n)=sum_{i=0,..n-1} (i!)^2. Note that 6 divides S(n) for n>1. For prime p=20879, p divides S(p-1). Hence p divides S(n) for all n >= p-1 and all prime values of S(n)/6 are for n < p-1.

The next term (a(7)) has 96 digits. The largest term (a(9)) has 288 digits. - Harvey P. Dale, Aug 31 2021

Harvey P. Dale, Table of n, a(n) for n = 1..9

Cf. A061062 (S(n)), A100288 (primes of the form S(n)-1), A100289 (n such that S(n)-1 is prime), A101747 (n such that S(n)/6 is prime).

f2=1; s=2; Do[f2=f2*n*n; s=s+f2; If[PrimeQ[s/6], Print[{n, s/6}]], {n, 2, 100}]
Select[Accumulate[(Range[0,25]!)^2]/6,PrimeQ] (* Harvey P. Dale, Aug 31 2021 *)

A101747 Numbers n such that ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 19, 40, 56, 93

Offset: 1

T. D. Noe, Dec 18 2004

Let S(n) = Sum_{i=0,..n-1} (i!)^2. Note that 6 divides S(n) for n>1. For prime p=20879, p divides S(p-1). Hence p divides S(n) for all n >= p-1 and all prime values of S(n)/6 are for n < p-1. These n yield provable primes for n <= 93. No other n < 4000.

No other n < 8000. [T. D. Noe, Jul 31 2008]

Cf. A061062 (S(n)), A100288 (primes of the form S(n)-1), A100289 (n such that S(n)-1 is prime), A101746 (primes of the form S(n)/6).

f2=1; s=2; Do[f2=f2*n*n; s=s+f2; If[PrimeQ[s/6], Print[{n, s/6}]], {n, 2, 100}]

A138586 a(1) = 1; a(n) = a(n-1) + (n!)^7.

Original entry on oeis.org

1, 129, 280065, 4586751489, 358322666751489, 100306488365546751489, 82606511560391889386751489, 173238283180457843219993066751489, 828593116199250458889895450218986751489

Offset: 1

Jonathan Vos Post, May 18 2008

After a(1) = 1 these are all divisible by 3. a(n)/3 is prime (i.e. a(n) is semiprime) for n = 2, 4 (i.e. (1!)^7 + (2!)^7 + (3!)^7 + (4!)^7 = 4586751489 = 3 * 1528917163) and then when next?

Cf. A000142, A001015, A104344, A100288.

a(n) = Sum_{k=1..n} (k!)^7 = Sum_{k=1..n} A001015(A000142(n)).

cp's OEIS Frontend

A061062 Sum of squared factorials: (0!)^2 + (1!)^2 + (2!)^2 + (3!)^2 +...+ (n!)^2.

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

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

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A101746 Primes of the form ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6.

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A101747 Numbers n such that ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6 is prime.

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A138586 a(1) = 1; a(n) = a(n-1) + (n!)^7.

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