A100289 -id:A100289 - cp's OEIS frontend

A001272 Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.

3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961

Offset: 1

At present the terms greater than or equal to 2653 are only probable primes.
Živković shows that all terms must be less than p = 3612703, which divides the alternating factorial af(k) for k >= p. - T. D. Noe, Jan 25 2008
Notwithstanding Živković's wording, p = 3612703 also divides the alternating factorial for k = 3612702. [Guy: If there is a value of k such that k + 1 divides af(k), then k + 1 will divide af(m) for all m > k.] Therefore af(3612701), approximately 7.3 * 10^22122513, is the final primality candidate. - Hans Havermann, Jun 17 2013
Next term (if it exists) has k > 100000 (per M. Rodenkirch post). - Eric W. Weisstein, Dec 18 2017

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 160, p. 52, Ellipses, Paris 2008.
Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, p. 198 (1997).
R. K. Guy, Unsolved Problems in Number Theory, B43.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 97.

factordb.com, Primality at factor-database of the alternating factorial for n=661.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
M. Rodenkirch, Alternating Factorials.
Eric Weisstein's World of Mathematics, Alternating Factorial.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Miodrag Živković, The number of primes Sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), no. 225, 403-409.
Index entries for sequences related to factorial numbers.

Cf. A005165, A002981, A002982, A100289.

with(numtheory); f := proc(n) local i; add((-1)^(n-i)*i!,i=1..n); end; isprime(f(15));

(* This program is not convenient for more than 15 terms *) Reap[For[n = 1, n <= 1000, n++, If[PrimeQ[Sum[(-1)^(n - k) * k!, {k, n}]], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Sep 05 2013 *)
Position[AlternatingFactorial[Range[200]], ?PrimeQ] // Flatten (* _Eric W. Weisstein, Sep 19 2017 *)

661 found independently by Eric W. Weisstein and Rachel Lewis (racheljlewis(AT)hotmail.com); 2653 and 3069 found independently by Chris Nash (nashc(AT)lexmark.com) and Rachel Lewis (racheljlewis(AT)hotmail.com)
3943, 4053, 4998 found by Paul Jobling (paul.jobling(AT)whitecross.com)
8275, 9158 found by team of Rachel Lewis, Paul Jobling and Chris Nash
661! - 660! + 659! - ... was shown to be prime by team of Giovanni La Barbera and others using the Certifix program developed by Marcel Martin, Jul 15 2000 (see link) - Paul Jobling (Paul.Jobling(AT)WhiteCross.com) and Giovanni La Barbera, Aug 02 2000
a(23) = 11164 found by Paul Jobling, Nov 25 2004
Edited by T. D. Noe, Oct 30 2008
Edited by Hans Havermann, Jun 17 2013
a(24) = 43592 from Serge Batalov, Jul 19 2017
a(25) = 59961 from Mark Rodenkirch, Sep 18 2017

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

Original entry on oeis.org

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

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).

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

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).

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}]

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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A001272 Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.

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

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

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

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

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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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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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