A038507 -id:A038507 - cp's OEIS frontend

A060371 a(n) = (prime(n) - 1)! + 1.

2, 3, 25, 721, 3628801, 479001601, 20922789888001, 6402373705728001, 1124000727777607680001, 304888344611713860501504000001, 265252859812191058636308480000001, 371993326789901217467999448150835200000001

Offset: 1

Jason Earls, Apr 01 2001

nonn

If the prime p is in A055469, that is if p = 2, 7, 11, 29, ... = A055469(j) which is valid for the first, 4th, 5th, 10th,.... entry here with j = 1, 2, 3, ..., then a(n) = A052295[A067186(j)] + 1. - R. J. Mathar, Apr 27 2007

It follows from Wilson's theorem that a(n) is divisible by the n-th prime. - Alonso del Arte, Feb 07 2014

Harry J. Smith, Table of n, a(n) for n = 1..88 (adapted by Vincenzo Librandi, Oct 17 2017)
Takashi Agoh, Karl Dilcher and Ladislav Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861. MR 98h:11003.
C. K. Caldwell, Wilson Primes
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449. MR 97c:11004.

Cf. A052295, A055469, A067186.

Subsequence of A038507. - Michel Marcus, Oct 17 2017

[Factorial(NthPrime(n)-1)+1: n in [1..15]]; // Vincenzo Librandi, Oct 17 2017

Table[(Prime[n] - 1)! + 1, {n, 12}] (* Alonso del Arte, Feb 07 2014 *)

{ n=1; forprime (p=1, 524, write("b060371.txt", n++, " ", (p - 1)! + 1); ) } \\ Harry J. Smith, Jul 04 2009

Corrected offset by Alonso del Arte, Feb 07 2014

A064237 Numbers k such that k! + 1 is divisible by a square.

Original entry on oeis.org

4, 5, 7, 12, 23

Offset: 1

Vladeta Jovovic, Sep 22 2001

229 is another term because 613^2 divides 229!+1. See A115091 for primes whose square divides m!+1 for some m. An examination of the factorizations of m!+1 for m<=100 found no additional squares. - T. D. Noe, Mar 01 2006
562 is also a term because 562!+1 is divisible by 563^2. - Vladeta Jovovic, Mar 30 2004
A web search reveals that for 1 <= k <= 228 there are 82 values of k for which k! + 1 has not been completely factored (the smallest is k=103), so showing that 229 and 562 are indeed the next two terms will be a huge task. I checked that k!+1 is not divisible by p^2 for k <= 1000 and prime p < 10^8. - Francois Brunault, Nov 23 2008
It is very likely that 229 and 562 are the next two terms, but this has not yet been proved. - Francois Brunault, Nov 29 2008
Contains A007540(n)-1 for all n.  That sequence is conjectured to be infinite. - Robert Israel, Jul 04 2016
This sequence includes A146968 (solutions of Brocard's problem). - Salvador Cerdá, Mar 08 2016
If k > 562 and k! + 1 is divisible by p^2 where p is prime, then either k > 10000 or p > 2038074743 (the hundred millionth prime). - Jason Zimba, Oct 21 2021

			4 is in the sequence because 4! + 1 = 5^2.
5 is in the sequence because 5! + 1 = 11^2.
6 is not in the sequence because 6! + 1 = 721
7 is in the sequence because 7! + 1 = 71^2.
12 is in the sequence because 12! + 1 = 13^2 * 2834329.
23 is a term because 23!+1 = 47^2*79*148139754736864591.
From _Thomas Richard_, Aug 31 2021: (Start)
229 and 562 are terms because
229!+1 = 613^2 * 38669 * 1685231 * 3011917759 * (417-digit composite)
562!+1 = 563^2 * 64467346976659839517037 * 112870688711507255213769871 * 63753966393108716329397432599379239 * (1214-digit prime). (End)

Hisanori Mishima, Factorizations of m!+1

Cf. A007540 (Wilson primes), A115091, A146968, A038507, A085692.

remove(t -> numtheory:-issqrfree(t!+1), [$1..50]); # Robert Israel, Jul 04 2016

Flatten[Position[MoebiusMu[Range[30]!+1], 0]]; (* T. D. Noe, Mar 01 2006, Nov 21 2008 *)

lista(nn) = for(n=1, nn, if(!issquarefree(n!+1), print1(n, ", "))); \\ Altug Alkan, Mar 08 2016

Example corrected by T. D. Noe, Nov 26 2008

A173319 a(n) = 5*n! + 1.

Original entry on oeis.org

6, 6, 11, 31, 121, 601, 3601, 25201, 201601, 1814401, 18144001, 199584001, 2395008001, 31135104001, 435891456001, 6538371840001, 104613949440001, 1778437140480001, 32011868528640001, 608225502044160001, 12164510040883200001, 255454710858547200001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), this sequence, A173314 (k=6).

[5*Factorial(n) + 1: n in [0..20]]; // Vincenzo Librandi, Sep 28 2013

[6] cat [n eq 1 select n+5 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 28 2013

A173319:=n->5*n!+1: seq(A173319(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[5 n! + 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 28 2013 *)

a(0)=6; for n>0, a(n) = n*a(n-1)-n+1. - Vincenzo Librandi, Sep 28 2013

(n-2)*a(n) - (n^2-n-1)*a(n-1) + (n-1)^2*a(n-2) = 0. [Bruno Berselli, Sep 28 2013]

A255342 Numbers n such that there are exactly two 1's in their factorial base representation (A007623).

Original entry on oeis.org

3, 7, 8, 11, 15, 21, 25, 26, 29, 30, 34, 37, 38, 41, 43, 44, 47, 51, 55, 56, 59, 63, 69, 75, 79, 80, 83, 87, 93, 99, 103, 104, 107, 111, 117, 121, 122, 125, 126, 130, 133, 134, 137, 139, 140, 143, 144, 148, 156, 160, 162, 166, 169, 170, 173, 174, 178, 181, 182, 185, 187, 188, 191, 193, 194, 197, 198, 202

Offset: 1

Antti Karttunen, Apr 27 2015

			The factorial base representation (A007623) of 3 is "11", which contains exactly two 1's, thus 3 is included in the sequence.
The f.b.r. of 7 is "101", with exactly two 1's, thus 7 is included in the sequence.
The f.b.r. of 21 is "311", with exactly two 1's, thus 21 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..13132

Cf. A007623, A257511, A255411, A255341, A255343.
Subsequence of A256450.
Subsequence: A038507 (apart from its initial 2 terms).

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 240}]; Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 2](* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)

A354891 a(n) = n! * Sum_{d|n} d^(n - d) / d!.

Original entry on oeis.org

1, 3, 7, 73, 121, 9721, 5041, 1760641, 44452801, 562615201, 39916801, 3156125575681, 6227020801, 192873372531841, 222245415808416001, 14806216643368550401, 355687428096001, 34884164976924636172801, 121645100408832001

Offset: 1

Seiichi Manyama, Jun 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..305

Cf. A038507, A342628, A354888, A354890, A354893.

a[n_] := n! * DivisorSum[n, #^(n - #)/#! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, d^(n-d)/d!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-(k*x)^k)))))

E.g.f.: Sum_{k>0} x^k/(k! * (1 - (k * x)^k)).

If p is prime, a(p) = 1 + p! = A038507(p).

A068481 Numbers k such that gcd(k!+1, 2^k+1) > 1.

Original entry on oeis.org

5, 9, 21, 33, 65, 81, 89, 113, 173, 209, 221, 245, 261, 281, 285, 309, 341, 345, 369, 393, 473, 509, 525, 545, 561, 593, 645, 725, 749, 785, 789, 833, 861, 873, 933, 953, 965, 1001, 1065, 1101, 1113, 1173, 1185, 1265, 1289, 1329, 1341, 1401, 1409, 1469

Offset: 1

Benoit Cloitre, Mar 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Cf. A000051 (2^n+1), A038507 (n!+1).

Cf. A068480, A068482, A068483.

Filtered([1..1470],n->Gcd(Factorial(n)+1,2^n+1)>1); # Muniru A Asiru, Oct 16 2018

select(n->gcd(factorial(n)+1,2^n+1)>1,[$1..1470]); # Muniru A Asiru, Oct 16 2018

Select[Range[2500], GCD[#! + 1, 2^# + 1] > 1 &] (* G. C. Greubel, Oct 15 2018 *)

isok(n) = gcd(n!+1,2^n+1) > 1; \\ Michel Marcus, Oct 16 2018

A073944 a(n) is the smallest m such that n-th prime divides m! + 1.

Original entry on oeis.org

1, 2, 4, 3, 5, 12, 16, 9, 14, 18, 30, 36, 40, 21, 23, 52, 15, 8, 18, 7, 72, 23, 13, 88, 96, 100, 6, 106, 86, 112, 63, 65, 16, 16, 50, 150, 156, 81, 166, 172, 89, 180, 95, 102, 196, 99, 210, 222, 61, 228, 64, 210, 240, 97, 31, 131, 9, 93, 40, 280, 282, 45, 63, 220, 312

Offset: 1

Jason Earls, Nov 13 2002

Essentially the same as A072937. - R. J. Mathar, Sep 23 2008

By Wilson's theorem, a(n) < prime(n). Sequence A115092 gives the number of m such that prime(n) divides m!+1. - T. D. Noe, Mar 01 2006, Jan 10 2009

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)

Cf. A038507.

Cf. A072937 (same sequence without a(1)).

Table[p=Prime[i]; m=1; While[m0, m++ ]; m, {i,100}] (* T. D. Noe, Mar 01 2006 *)
Module[{sm=Table[{m,m!+1},{m,400}]},Table[SelectFirst[sm,Mod[#[[2]],p]==0&],{p,Prime[ Range[70]]}]][[;;,1]] (* Harvey P. Dale, Sep 15 2023 *)

A093804 Primes p such that p! + 1 is also prime.

Original entry on oeis.org

2, 3, 11, 37, 41, 73, 26951, 110059, 150209

Offset: 1

Jason Earls, May 19 2004

Or, numbers n such that Sum_{d|n} d! is prime.
The prime 26951 from A002981 (n!+1 is prime) is a member since Sum_{d|n} d! = n!+1 if n is prime. - Jonathan Sondow, Jan 30 2005
a(n) are the primes in A002981[n] = {0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ...} Numbers n such that n! + 1 is prime. Corresponding primes of the form p! + 1 are listed in A103319[n] = {3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, ...}. - Alexander Adamchuk, Sep 23 2006

			Sum_{d|3} d! = 1! + 3! = 7 is prime, so 3 is a member.

Chris K. Caldwell, The List of Largest Known Primes, 110059! + 1
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
Apoloniusz Tyszka, A common approach to the problem of the infinitude of twin primes, primes of the form n! + 1, and primes of the form n! - 1, 2018.
Apoloniusz Tyszka, A new approach to solving number theoretic problems, 2018.

Cf. A062363, A002981, A038507, A088332, A103317, A103319.

seq(`if`(isprime(ithprime(n)!+1), ithprime(n), NULL),n=1..25); # Nathaniel Johnston, Jun 28 2011

Select[Prime[Range[5! ]],PrimeQ[ #!+1]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

isok(n) = ispseudoprime(n) && ispseudoprime(n!+1); \\ Jinyuan Wang, Jan 20 2020

One more term from Alexander Adamchuk, Sep 23 2006
a(8)=110059 (found on Jun 11 2011, by PrimeGrid), added by Arkadiusz Wesolowski, Jun 28 2011
a(9)=150209 (found on Jun 09 2012, by Rene Dohmen), added by Jinyuan Wang, Jan 20 2020

A116893 Numbers k such that gcd(k!+1, k^k+1) > 1.

Original entry on oeis.org

1, 3, 23, 39, 51, 63, 95, 99, 131, 183, 191, 215, 239, 251, 299, 303, 315, 363, 371, 411, 419, 431, 443, 495, 543, 575, 659, 683, 711, 743, 755, 791, 831, 891, 911, 935, 975, 1019, 1031, 1055, 1071, 1143, 1155, 1191, 1211, 1223, 1251, 1275, 1295, 1355

Offset: 1

Giovanni Resta, Mar 01 2006

See A116892 for the corresponding values of the GCD. See also comments in A116891.

			gcd(1!+1, 1^1+1) = 2, gcd(2!+1, 2^2+1) = 1 and gcd(3!+1, 3^3+1) = 7, so 1 and 3 are the first two terms of the sequence.

Nick Hobson, Table of n, a(n) for n = 1..10000 (first 1832 terms from Antti Karttunen)
Nick Hobson, C program.

Cf. A014566, A038507, A067658, A116891, A116892, A116894.

C
```
See Links section.
```

Select[Range[1500], (GCD[ #!+1, #^#+1] > 1)&]

isok(n) = gcd(n! + 1, n^n + 1) != 1; \\ Michel Marcus, Jul 22 2018

A216071 Brocard's problem: positive integers m such that m^2 = n! + 1 for some n.

Original entry on oeis.org

5, 11, 71

Offset: 1

V. Raman, Sep 01 2012

See A085692 and A146968 for links, references and comments. - M. F. Hasler, Nov 20 2018

Wikipedia, Brocard's problem

A085692, A146968, A216071 are all essentially the same sequence. - N. J. A. Sloane, Sep 01 2012

Sqrt[#!+1]&/@Select[Range[1000],IntegerQ[Sqrt[#!+1]]&] (* Harvey P. Dale, Sep 29 2012 *)

apply( sqrtint, A085692) \\ M. F. Hasler, Nov 20 2018

select( is_A216071(m)=m^2==A084558(m^2)!+1, [0..99]) \\ M. F. Hasler, Nov 20 2018

a(n) = A000196(A085692(n)) = A000196(A038507(A146968(n))) where A000196 = sqrt and A038507(n) = n! + 1. - M. F. Hasler, Nov 20 2018

cp's OEIS Frontend

A060371 a(n) = (prime(n) - 1)! + 1.

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Magma

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A064237 Numbers k such that k! + 1 is divisible by a square.

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Maple

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A173319 a(n) = 5*n! + 1.

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Magma

Magma

Maple

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A255342 Numbers n such that there are exactly two 1's in their factorial base representation (A007623).

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Mathematica

A354891 a(n) = n! * Sum_{d|n} d^(n - d) / d!.

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Mathematica

PARI

PARI

Formula

A068481 Numbers k such that gcd(k!+1, 2^k+1) > 1.

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GAP

Maple

Mathematica

PARI

A073944 a(n) is the smallest m such that n-th prime divides m! + 1.

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Mathematica

A093804 Primes p such that p! + 1 is also prime.

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