A061062 -id:A061062 - 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

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

A100288 Primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2.

Original entry on oeis.org

5, 41, 617, 15017, 25935017, 1651637417, 13301522971817, 41117342095090841723228045851817, 2616218222822143606864564493635469851817

Offset: 1

T. D. Noe, Nov 11 2004

Jonathan Vos Post contributed these numbers to Prime Curios.

			41 = (1!)^2 + (2!)^2 + (3!)^2 is prime.

G. L. Honaker, Jr. and C. Caldwell, Prime Curios!

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

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 *)

A289948 a(n) = Sum_{k=0..n} k!^3.

Original entry on oeis.org

1, 2, 10, 226, 14050, 1742050, 374990050, 128399054050, 65676719822050, 47850402559694050, 47832576242431694050, 63649302669112063694050, 109966989623147836159694050, 241567605673714904675071694050, 662801328154821495670649599694050, 2236801993181528581580834681599694050

Offset: 0

Seiichi Manyama, Jul 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..181

Cf. A003422(n+1) (k!), A061062 (k!^2), this sequence (k!^3), A289949 (k!^4).

With[{nn = 16}, Table[Total@ Take[#, n], {n, nn}] &@ Table[k!^3, {k, 0, nn}]] (* Michael De Vlieger, Jul 16 2017 *)
Accumulate[(Range[0,20]!)^3] (* Harvey P. Dale, Nov 30 2017 *)

a(n) = sum(k=0, n, k!^3); \\ Michel Marcus, Jul 16 2017

A289949 a(n) = Sum_{k=0..n} k!^4.

Original entry on oeis.org

1, 2, 18, 1314, 333090, 207693090, 268946253090, 645510228813090, 2643553803594573090, 17342764866576345933090, 173418555892594089945933090, 2538940579958951120707545933090, 52646414799433780559063261145933090, 1503614384819523432725006336630745933090

Offset: 0

Seiichi Manyama, Jul 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..144

Cf. A289945.

Sum_{k=0..n} k!^m: A003422(n+1) (m=1), A061062 (m=2), A289948 (m=3), this sequence (m=4).

Accumulate[(Range[0,15]!)^4] (* Harvey P. Dale, Nov 29 2020 *)

a(n) = sum(k=0, n, k!^4); \\ Michel Marcus, Jul 16 2017

a(n) = A289945(n) + 1 for n > 0.

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}]

A173476 Triangle T(n, k) = 1 + (k!)^2 - 2k!(n-k)! + ((n-k)!)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 26, 2, 2, 26, 530, 26, 1, 26, 530, 14162, 530, 17, 17, 530, 14162, 516962, 14162, 485, 1, 485, 14162, 516962, 25391522, 516962, 13925, 325, 325, 13925, 516962, 25391522, 1625621762, 25391522, 515525, 12997, 1, 12997, 515525, 25391522, 1625621762

Offset: 0

Roger L. Bagula, Feb 19 2010

			Triangle begins as:
           1;
           1,        1;
           2,        1,      2;
          26,        2,      2,    26;
         530,       26,      1,    26, 530;
       14162,      530,     17,    17, 530, 14162;
      516962,    14162,    485,     1, 485, 14162, 516962;
    25391522,   516962,  13925,   325, 325, 13925, 516962, 25391522;
  1625621762, 25391522, 515525, 12997,   1, 12997, 515525, 25391522, 1625621762;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A003149, A061062.

[(Factorial(n-k) -Factorial(k))^2 +1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021

Table[((n-k)! -k!)^2 +1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)

flatten([[(factorial(n-k) -factorial(k))^2 +1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021

T(n, k) = 1 + ( (n-k)! - k! )^2.

Sum_{k=0..n} T(n, k) = 1 + n + 2*A061062(n) - 2*A003149(n). - G. C. Greubel, Feb 19 2021

Edited by G. C. Greubel, Feb 19 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A100288 Primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A289948 a(n) = Sum_{k=0..n} k!^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A289949 a(n) = Sum_{k=0..n} k!^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A173476 Triangle T(n, k) = 1 + (k!)^2 - 2*k!*(n-k)! + ((n-k)!)^2, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

A173476 Triangle T(n, k) = 1 + (k!)^2 - 2k!(n-k)! + ((n-k)!)^2, read by rows.