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Search: a100289 -id:a100289
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A001272
Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.
+10
10
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
REFERENCES
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.
LINKS
Table of n, a(n) for n=1..25.
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.
MAPLE
with(numtheory); f := proc(n) local i; add((-1)^(n-i)*i!, i=1..n); end; isprime(f(15));
MATHEMATICA
(* 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 *)
CROSSREFS
Cf. A005165, A002981, A002982, A100289.
KEYWORD
nonn,hard,more,nice,fini
AUTHOR
N. J. A. Sloane
EXTENSIONS
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
STATUS
approved
A104344
a(n) = Sum_{k=1..n} k!^2.
+10
10
1, 5, 41, 617, 15017, 533417, 25935017, 1651637417, 133333531817, 13301522971817, 1606652445211817, 231049185247771817, 39006837228880411817, 7639061293780877851817, 1717651314017980301851817, 439480788011413032845851817, 126953027293558583218061851817
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Factorial Sums
FORMULA
a(n) = A061062(n) - 1. - Michel Marcus, Feb 28 2014
MATHEMATICA
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 *)
PROG
(PARI) a(n) = sum(k=1, n, k!^2); \\ Michel Marcus, Jul 16 2017
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 02 2005
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Sep 24 2009
STATUS
approved
A100288
Primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2.
+10
6
5, 41, 617, 15017, 25935017, 1651637417, 13301522971817, 41117342095090841723228045851817, 2616218222822143606864564493635469851817
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Jonathan Vos Post contributed these numbers to Prime Curios.
LINKS
Table of n, a(n) for n=1..9.
G. L. Honaker, Jr. and C. Caldwell, Prime Curios!
EXAMPLE
41 = (1!)^2 + (2!)^2 + (3!)^2 is prime.
PROG
(PARI) lista(nn) = {my(s=1); for(k=2, nn, s+=(k!)^2; if(ispseudoprime(s), print1(s, ", "))); } \\ Jinyuan Wang, Mar 08 2020
CROSSREFS
Cf. A100289 (k such that (1!)^2 + (2!)^2 + (3!)^2 + ... + (k!)^2 is prime).
See also A061062, A104344.
KEYWORD
nonn,fini
AUTHOR
T. D. Noe, Nov 11 2004
STATUS
approved
A289947
Values of n for which Sum_{k=1..n} k!^6 is prime.
+10
3
5, 34, 102
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..3.
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Integer Sequence Primes
EXAMPLE
A289946(5) = 2986175149697 is prime.
PROG
(PARI) isok(n) = isprime(sum(k=1, n, k!^6)); \\ Michel Marcus, Jul 17 2017
CROSSREFS
Cf. A289946 (Sum_{k=1..n} k!^6).
Cf. A100289 (k!^2), A290014 (k!^10).
KEYWORD
nonn,bref,full,fini
AUTHOR
Eric W. Weisstein, Jul 16 2017
STATUS
approved
A290014
Values of n for which Sum_{k=1..n} k!^10 is prime.
+10
3
3, 4, 5, 16, 25
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..5.
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Integer Sequence Primes
EXAMPLE
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.
...
CROSSREFS
Cf. A100289 (k!^2), A289947 (k!^6).
KEYWORD
nonn,bref,full,fini
AUTHOR
Eric W. Weisstein, Jul 17 2017
STATUS
approved
A101746
Primes of the form ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6.
+10
2
7, 103, 2503, 88903, 4322503, 2473107965928318342544472044975303
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..9
MATHEMATICA
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 *)
CROSSREFS
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).
KEYWORD
fini,nonn
AUTHOR
T. D. Noe, Dec 18 2004
STATUS
approved
A101747
Numbers n such that ((0!)^2+(1!)^2+(2!)^2+...+(n!)^2)/6 is prime.
+10
1
3, 4, 5, 6, 7, 19, 40, 56, 93
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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]
LINKS
Table of n, a(n) for n=1..9.
MATHEMATICA
f2=1; s=2; Do[f2=f2*n*n; s=s+f2; If[PrimeQ[s/6], Print[{n, s/6}]], {n, 2, 100}]
CROSSREFS
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).
KEYWORD
fini,nonn
AUTHOR
T. D. Noe, Dec 18 2004
STATUS
approved
A290250
Smallest (prime) number a(n) > 2 such that Sum_{k=1..a(n)} k!^(2*n) is divisible by a(n).
+10
0
1248829, 13, 1091, 13, 41, 37, 463, 13, 23, 13, 1667, 37, 23, 13, 41, 13, 139
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..17.
Eric Weisstein's World of Mathematics, Factorial Sums
EXAMPLE
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
MATHEMATICA
Table[Module[{sum = 1, fac = 1, k = 2}, While[! Divisible[sum += (fac *= k)^(2 n), k], k++]; k], {n, 17}]
CROSSREFS
Cf. A100289 (n such that Sum_{k=1..n} k!^2 is prime), A289945 (k!^4), A289946 (k!^6), A290014 (k!^10).
KEYWORD
nonn,more,hard
AUTHOR
Eric W. Weisstein, Jul 24 2017
STATUS
approved
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