A002981 -id:A002981 - cp's OEIS frontend

A002583 Largest prime factor of n! + 1.

2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591, 765041185860961084291, 38681321803817920159601

Offset: 0

N. J. A. Sloane

Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1.
Cf. Wilson's theorem (1770): p | (p-1)! + 1 iff p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			(0!+1)=[2], (1!+1)=[2], (2!+1)=[3], (3!+1)=[7], (4!+1)=25=5*[5], (5!+1)=121=11*[11], (6!+1)=721=7*[103], (7!+1)=5041=71*[71], etc. - Mitch Cervinka (puritan(AT)toast.net), May 11 2009

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 0..139 (first 101 terms originally derived from Hisanori Mishima's data by T. D. Noe)
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Li Lai, On the largest prime divisor of n! + 1, arXiv:2103.14894 [math.NT], 2021.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Blake C. Stacey, Equiangular Lines, Ch. 1, A First Course in the Sporadic SICs, SpringerBriefs in Math. Phys. (2021) Vol. 41, see page 5.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A002981, A038507, A051301, A056111, A096225.

[Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020

PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n!+1][[ -1]],{n,0,35}] ..and/or.. Table[FactorInteger[n!+1,FactorComplete->True][[ -1,1]],{n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
FactorInteger[#][[-1,1]]&/@(Range[0,30]!+1) (* Harvey P. Dale, Sep 04 2017 *)

a(n)=my(f=factor(n!+1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (1-o(1))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012

Lai proves that lim sup a(n)/n > 7.238. - Charles R Greathouse IV, Jun 22 2021

More terms from Robert G. Wilson v, Aug 01 2000

Corrected by Jud McCranie, Jan 03 2001

A204657 Numbers n such that n!10 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 13, 19, 21, 25, 41, 57, 79, 127, 135, 149, 165, 177, 193, 209, 223, 255, 273, 287, 297, 375, 433, 459, 481, 565, 1079, 1435, 1543, 1771, 1913, 1983, 2063, 2305, 2653, 6789, 8757, 11149, 13671, 15433, 16369, 17261, 18129, 22129, 22785, 22875, 25235, 25247, 26329, 27675, 33391, 39075, 41195, 47435, 47621, 48409, 59235, 59715, 61571, 65433, 78761, 83033

Offset: 1

M. F. Hasler, Jan 17 2012

n!10 = Product_{k=0..floor((n-1)/10)}(n - 10k).
a(61) > 50000. - Robert Price, Jun 10 2012
The first 11 primes associated with this sequence: 3, 3, 5, 7, 11, 13, 41, 173, 233, 1877, 293603. - Robert Price, Mar 10 2017
a(67) > 10^5. - Robert Price, Mar 31 2017

C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.
Nathan Russell, n!10+2 results, primenumbers group, Jan 2012.
Nathan Russell, n!10+2 results, message 23995 in primenumbers Yahoo group, Jan 17, 2012.

Cf. A204658, A204659, A204660, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 10] + 2] &]

for(n=0,9999,isprime(prod(i=0,(n-2)\10,n-10*i)+2)& print1(n","))

a(40)-a(59) from Robert Price, Jun 10 2012
Inserted missing term of 6789 by Robert Price, Mar 10 2017
a(61)-a(66) from Robert Price, Mar 31 2017

A051301 Smallest prime factor of n!+1.

Original entry on oeis.org

2, 2, 3, 7, 5, 11, 7, 71, 61, 19, 11, 39916801, 13, 83, 23, 59, 17, 661, 19, 71, 20639383, 43, 23, 47, 811, 401, 1697, 10888869450418352160768000001, 29, 14557, 31, 257, 2281, 67, 67411, 137, 37, 13763753091226345046315979581580902400000001

Offset: 0

Labos Elemer

nonn

Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N! + 1.
Cf. Wilson's Theorem (1770): p | (p-1)! + 1 if and only if p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			a(3) = 7 because 3! + 1 = 7.
a(4) = 5 because 4! + 1 = 25 = 5^2. (5! + 1 is also the square of a prime).
a(6) = 7 because 6! + 1 = 721 = 7 * 103.

Albert H. Beiler, "Recreations in The Theory of Numbers, The Queen of Mathematics Entertains," Dover Publ. NY, 1966, Page 49.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).

Chai Wah Wu, Table of n, a(n) for n = 0..138 n = 0..100 derived from Hisanori Mishima's data by T. D. Noe.
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002583, A002981, A038507, A096225.

with(numtheory): A051301 := n -> sort(convert(divisors(n!+1),list))[2]; # Corrected by Peter Luschny, Jul 17 2009

Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ]
FactorInteger[#][[1,1]]&/@(Range[0,40]!+1) (* Harvey P. Dale, Oct 16 2021 *)

a(n)=factor(n!+1)[1,1] \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n + 1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

By Wilson's theorem, a(n) >= n + 1 with equality if and only if n + 1 is prime. - Chai Wah Wu, Jul 14 2019

A051915 Numbers k such that 2*k! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 12, 18, 35, 51, 53, 78, 209, 396, 4166, 9091, 9587, 13357, 15917, 17652, 46127, 66480

Offset: 1

Labos Elemer, Dec 18 1999

Used PrimeForm to prove primality for n = 4166 (classical N-1 test). - David Radcliffe, May 28 2007

a(22) > 80000. - Serge Batalov, Jun 09 2025

			k = 5 is here because 2*5! + 1 = 241 is prime.

Cf. A002981, A076679, A076680, A076681, A076682, A076683, A178488, A180626, A126896.

[n: n in [0..1000] | IsPrime(2*Factorial(n) +1)]; // Vincenzo Librandi, Feb 21 2015

Select[Range[0, 400], PrimeQ[2*#! + 1] &] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

is(k) = ispseudoprime(2*k!+1); \\ Jinyuan Wang, Feb 05 2020

4166 from David Radcliffe, May 28 2007
More terms from Serge Batalov, Feb 18 2015
a(21) from Serge Batalov, Jun 08 2025

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)))

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

A156165 Numbers k such that k![7]+1 is prime (n![7] = A114799(n) = septuple factorial).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 10, 12, 13, 24, 25, 26, 29, 31, 35, 36, 47, 49, 57, 58, 64, 71, 73, 75, 78, 80, 97, 123, 125, 129, 131, 135, 147, 150, 159, 183, 201, 250, 251, 255, 298, 336, 337, 458, 467, 556, 570, 657, 743, 801, 908, 925, 1003, 1209, 1473, 1524, 1716, 1881, 1926

Offset: 1

M. F. Hasler, Feb 10 2009

nonn

a(103) > 50000. - Robert Price, Sep 03 2012

Robert Price, Table of n, a(n) for n = 1..102
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!7+1.

Cf. A156167, A085150, A085148, A085146, A037083, A080778, A002981.

mf[n_, k_] := Product[n - i k, {i, 0, Quotient[n - 2, k]}];
Reap[For[k = 0, k <= 2000, k++, If[PrimeQ[mf[k, 7] + 1], Sow[k]]]][[2, 1]] (* Jean-François Alcover, Feb 26 2019 *)
Select[Range[0,2000],PrimeQ[Times@@Range[#,1,-7]+1]&] (* Harvey P. Dale, Aug 21 2021 *)

mf(n,k=7)=prod(i=0,(n-2)\k,n-i*k)
for( n=0,9999, ispseudoprime(mf(n)+1) & print1(n","))

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

A076679 Numbers k such that 3*k! + 1 is prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 13, 23, 25, 32, 38, 40, 47, 96, 3442, 4048, 4522, 4887, 7033, 9528, 12915, 31762, 114482

Offset: 1

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

a(25) > 115000. - Serge Batalov, Jun 15 2025

			k = 6 is here because 3*6! + 1 = 2161 is prime.

Cf. A002981, A051915, A076680, A076681, A076682, A076683, A178488, A180626, A126896.

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

ABC2 3*$a!+1
a: from 1 to 1000 // Jinyuan Wang, Feb 05 2020

More terms from Serge Batalov, Feb 18 2015
a(20)-a(23) from Roger Karpin, Nov 13 2016
a(24) from Serge Batalov, Jun 15 2025

A126896 Numbers k such that 10*k! + 1 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 23, 32, 39, 61, 349, 718, 805, 1025, 1194, 1550, 1774, 3417, 7583

Offset: 1

Parthasarathy Nambi, May 07 2007

a(17) > 2880. - Jinyuan Wang, Feb 05 2020

a(19) > 12000. - Michael S. Branicky, Jul 07 2024

			k = 4 is a term because 10*4! + 1 = 241 is prime.

Corresponding primes are in A089764.

Cf. A002981, A051915, A076679, A076680, A076681, A076682, A076683, A178488, A180626.

is(k) = ispseudoprime(10*k!+1); \\ Jinyuan Wang, Feb 05 2020

a(15)-a(16) from Jinyuan Wang, Feb 05 2020
a(17) from Michael S. Branicky, Apr 16 2023
a(18) from Michael S. Branicky, Jul 07 2024

cp's OEIS Frontend

A002583 Largest prime factor of n! + 1.

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A204657 Numbers n such that n!10 + 2 is prime.

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A051301 Smallest prime factor of n!+1.

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A051915 Numbers k such that 2*k! + 1 is prime.

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Magma

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

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Mathematica

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

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A156165 Numbers k such that k![7]+1 is prime (n![7] = A114799(n) = septuple factorial).

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