A002982 -id:A002982 - cp's OEIS frontend

A151913 Numbers n for which (8+n!)/8 is prime.

7, 9, 10, 12, 14, 20, 23, 24, 29, 44, 108, 2049, 3072, 4862, 8807, 15089

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

a(17) > 25000. - Robert Price, Dec 20 2016

For primes of the form (8+k!!)/8 see A139066.
Cf. A089130, A117141, A139035, A139057, A139059, A139060, A139062, A139064, A139067.
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

a = {}; Do[If[PrimeQ[(n! + 8)/8], AppendTo[a, n]], {n, 1, 500}]; a

is(n)=n>6 && isprime((8+n!)/8) \\ Charles R Greathouse IV, Apr 29 2016

Definition corrected Feb 24 2010
More terms from Serge Batalov, Feb 18 2015
a(15)-a(16) from Robert Price, Dec 20 2016

A055490 Factorial primes: primes of the form n! - 1.

Original entry on oeis.org

5, 23, 719, 5039, 479001599, 87178291199, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 1

Labos Elemer, Jun 28 2000

nonn

For further information see A002982, which is the main entry.
Also primes of the form 1*1! + 2*2! + ... + n*n!. - Jonathan Vos Post, Jul 21 2006
Prime numbers that are the difference of two factorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.

T. D. Noe, Table of n, a(n) for n=1..12

Cf. A000142, A000203, A002982.

Select[Range[40]!-1,PrimeQ] (* Harvey P. Dale, Aug 16 2012 *)

p = n!-1 for some n given in A002982.

Edited by Jon E. Schoenfield, Jan 09 2015

A200906 Numbers n such that cyclotomic polynomial value Phi(5,n!) is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 21, 44, 64, 244, 268, 2415

Offset: 1

Serge Batalov, Nov 23 2011

nonn

2415 corresponds to a probable prime. - Serge Batalov, Nov 24 2011

a(12) > 15000. - Robert Price, Jun 20 2015

			5 is in the sequence because Phi(5,5!) = ((5!)^5-1)/(5!-1)= 209102521 is prime.

Cf. A002981, A002982, A046029, A046029.

Do[If[PrimeQ[Cyclotomic[5, n!]], Print[n]], {n, 0, 600}]

for(n=0,600,x=n!;if(isprime(eval(polcyclo(5))),print(n)))

A335407 Number of anti-run permutations of the prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A076680 Numbers k such that 4*k! + 1 is prime.

Original entry on oeis.org

0, 1, 4, 7, 8, 9, 13, 16, 28, 54, 86, 129, 190, 351, 466, 697, 938, 1510, 2748, 2878, 3396, 4057, 4384, 5534, 7069, 10364

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 25 2002

a(25) > 6311. - Jinyuan Wang, Feb 06 2020

			k = 7 is a term because 4*7! + 1 = 20161 is prime.

Factor Database, Factors of the numbers 4z!+1

Cf. m*n!-1 is a prime: A076133, A076134, A099350, A099351, A180627-A180631.
Cf. m*n!+1 is a prime: A051915, A076679-A076683, A178488, A180626, A126896.
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205.
Cf. n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071.

Select[Range[5000],PrimeQ[4#!+1]&] (* Harvey P. Dale, Mar 23 2011 *)

is(k) = ispseudoprime(4*k!+1); \\ Jinyuan Wang, Feb 06 2020

Corrected (added missed terms 2748, 2878) by Serge Batalov, Feb 24 2015
a(24) from Jinyuan Wang, Feb 06 2020
a(25)-a(26) from Michael S. Branicky, Jul 04 2024

A076133 Numbers k such that 2*k! - 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 14, 15, 17, 22, 28, 91, 253, 257, 298, 659, 832, 866, 1849, 2495, 2716, 2773, 2831, 3364, 5264, 7429, 28539, 32123, 37868, 65591, 113920

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 30 2002

a(32) > 116000. - Serge Batalov, Jun 06 2025

			k = 5 is here because 2*5! - 1 = 239 is prime.

Cf. A051915.

Cf. A002982, A076134, A099350, A099351, A180627, A180628, A180629, A180630, A180631.

[n: n in [1..600] | IsPrime(2*Factorial(n)-1)]; // Vincenzo Librandi, Feb 20 2015

Select[Range[8000], PrimeQ[2 #! - 1] &] (* Vincenzo Librandi, Feb 20 2015 *)

is(k) = ispseudoprime(2*k!-1); \\ Jinyuan Wang, Feb 04 2020

a(24)-a(29) from Serge Batalov, Feb 18 2015
a(30) from Serge Batalov, Jun 03 2025
a(31) from Serge Batalov, Jun 06 2025

A076134 Numbers k such that 3*k! - 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 12, 17, 26, 76, 379, 438, 1695, 6709, 13313, 18504, 19021, 24488, 45552, 49085, 65451

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 30 2002

a(23) > 80000. - Serge Batalov, Jun 09 2025

			k = 5 is here because 3*5! - 1 = 359 is prime.

Cf. A051915, A076134.

Cf. A002982, A076133, A099350, A099351, A180627, A180628, A180629, A180630, A180631.

for n from 0 to 1000 do if isprime(3*n! - 1) then print(n) end if end do;

Select[Range[0, 10^3], PrimeQ[3 #! - 1] &] (* Robert Price, May 27 2019 *)

isok(n) = isprime(3*n! - 1); \\ Michel Marcus, Nov 13 2016

ABC2 3*$a!+1
a: from 1 to 1000 // Jinyuan Wang, Feb 04 2020

a(15)-a(21) from Roger Karpin, Nov 13 2016

a(22) from Serge Batalov, Jun 08 2025

A099350 Numbers k such that 4*k! - 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 10, 11, 51, 63, 197, 313, 579, 1264, 2276, 2669, 4316, 4382, 4678, 7907, 10843

Offset: 1

Brian Kell, Oct 12 2004

a(19) > 4570. - Jinyuan Wang, Feb 04 2020

			k = 5 is here because 4*5! - 1 = 479 is prime.

Cf. A076680.

Cf. A002982, A076133, A076134, A099351, A180627, A180628, A180629, A180630, A180631.

for n from 0 to 1000 do if isprime(4*n! - 1) then print(n) end if end do;

For[n = 0, True, n++, If[PrimeQ[4 n! - 1], Print[n]]] (* Jean-François Alcover, Sep 23 2015 *)

is_A099350(n)=ispseudoprime(n!*4-1) \\ M. F. Hasler, Sep 20 2015

a(14) from Alois P. Heinz, Sep 21 2015
a(15)-a(16) from Jean-François Alcover, Sep 23 2015
a(17)-a(18) from Jinyuan Wang, Feb 04 2020
a(19) from Michael S. Branicky, May 16 2023
a(20)-a(21) from Michael S. Branicky, Jul 11 2024

A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

Original entry on oeis.org

1, 4, 11, 18, 24, 31, 37, 44, 45, 50, 52, 57, 58, 65, 66, 70, 71, 73, 76, 78, 79, 83, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 107, 108, 109, 110, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 131

Offset: 1

Gus Wiseman, May 13 2019

nonn

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Conjecture: 137 is the greatest integer not in this sequence. - Charlie Neder, May 14 2019

			The sequence of terms together with an integer partition of each whose reciprocal factorial sum is 1 begins:
   1: (1)
   4: (2,2)
  11: (3,3,3,2)
  18: (3,3,3,3,3,3)
  24: (4,4,4,4,3,3,2)
  31: (4,4,4,4,3,3,3,3,3)
  37: (4,4,4,4,4,4,4,4,3,2)
  44: (4,4,4,4,4,4,4,4,3,3,3,3)
  45: (5,5,5,5,5,4,4,4,3,3,2)
  50: (4,4,4,4,4,4,4,4,4,4,4,4,2)

Charlie Neder, Table of n, a(n) for n = 1..1000

Factorial numbers: A000142, A002982, A007489, A011371, A022559, A064986, A115627, A284605, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316855, A325619, A325620, A325621, A325622, A325623, A325624.

a(11)-a(55) from Charlie Neder, May 14 2019

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

Original entry on oeis.org

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

N. J. A. Sloane

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

cp's OEIS Frontend

A151913 Numbers n for which (8+n!)/8 is prime.

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A055490 Factorial primes: primes of the form n! - 1.

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A200906 Numbers n such that cyclotomic polynomial value Phi(5,n!) is prime.

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A335407 Number of anti-run permutations of the prime indices of n!.

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A076680 Numbers k such that 4*k! + 1 is prime.

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A076133 Numbers k such that 2*k! - 1 is prime.

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A076134 Numbers k such that 3*k! - 1 is prime.

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