A117141 -id:A117141 - cp's OEIS frontend

A007749 Numbers k such that k!! - 1 is prime.

3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318, 76190, 100654, 145706

Offset: 1

N. J. A. Sloane

a(n) is even for n>1. a(n) = 2*A091415(n-1) for n>1, where A091415(n) = {2, 3, 4, 8, 13, 32, 41, 45, 59, 97, 107, 364, 421, 444, 1164, 1738, 3202, 4335, 4841, ...} (numbers k such that k!*2^k - 1 is prime). Corresponding primes of the form k!!-1 are listed in A117141 = {2, 7, 47, 383, 10321919, 51011754393599, ...}. - Alexander Adamchuk, Nov 19 2006

The PFGW program has been used to certify all the terms up to a(25), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Apr 22 2016

The Top Ten (a Catalogue of Primal Configurations) from the unpublished collections of R. Ondrejka, assisted by C. Caldwell and H. Dubner, March 11, 2000, Page 61.

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!2-1.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations
Eric Weisstein's World of Mathematics, Double Factorial Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Index entries for sequences related to factorial numbers

Cf. A006882.

Cf. A091415 (n such that n!*2^n - 1 is prime), A117141 (primes of the form n!! - 1).

select(t -> isprime(doublefactorial(t)-1), [3, seq(n,n=4..3000,2)]); # Robert Israel, Apr 21 2016

a(1) = 3, for n>1 k=2;f=2;Do[k=k+2;f=f*k;If[PrimeQ[f-1],Print[k]],{n,2,5000}] (* Alexander Adamchuk, Nov 19 2006 *)
Select[Range[45000],PrimeQ[#!!-1]&] (* Harvey P. Dale, Aug 07 2013 *)

print1(3);for(n=2, 1e3, if(ispseudoprime(n!<Charles R Greathouse IV, Jun 16 2011

a(n) = 2*A091415(n-1) for n>1. - Alexander Adamchuk, Nov 19 2006

Entry updated by Robert G. Wilson v, Aug 18 2000
Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
a(23)-a(24) from Sou Fukui, Jun 05 2015
a(25) from Sou Fukui, Apr 21 2016

A139056 Numbers k for which (k!-3)/3 is prime.

Original entry on oeis.org

4, 6, 12, 16, 29, 34, 43, 111, 137, 181, 528, 2685, 39477, 43697

Offset: 1

Artur Jasinski, Apr 07 2008

Corresponding primes (k!-3)/3 are in A139057.
a(13) > 10000. The PFGW program has been used to certify all the terms up to a(12), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Mar 28 2014
98166 is a member of the sequence but its index is not yet determined. The interval where sieving and tests were not run is [60000,90000]. - Serge Batalov, Feb 24 2015

C. Caldwell. The Prime database entry for the prime generated by a(i)=98166.

Cf. A007749, A117141.
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.
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.

a = {}; Do[If[PrimeQ[(-3 + n!)/3], AppendTo[a, n]], {n, 1, 1000}]; a

for(n=1,1000,if(floor(n!/3-1)==n!/3-1,if(ispseudoprime(n!/3-1),print(n)))) \\ Derek Orr, Mar 28 2014

Definition corrected by Derek Orr, Mar 28 2014
a(8)-a(11) from Derek Orr, Mar 28 2014
a(12) from Giovanni Resta, Mar 28 2014
a(13)-a(14) from Serge Batalov, Feb 24 2015

A139066 Primes of the form (8+k!)/8.

Original entry on oeis.org

631, 45361, 453601, 59875201, 10897286401, 304112751022080001, 3231502092360622080001, 77556050216654929920001, 1105220249217462744317952000001, 332283946848556096005453226376826986289954816000000001

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For numbers k for which (8+k!)/8 is prime see A151913.

The next term (a(11)) has 174 digits. - Harvey P. Dale, May 10 2016

Amiram Eldar, Table of n, a(n) for n = 1..11

Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139155, A151913.

a = {}; Do[If[PrimeQ[(n! + 8)/8], AppendTo[a, (n! + 8)/8]], {n, 1, 50}]; a
Select[(8+Range[50]!)/8,PrimeQ] (* Harvey P. Dale, May 10 2016 *)

for(k=4,1e3,if(ispseudoprime(t=k!/8+1),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A139155(A151913(n)). - Amiram Eldar, Oct 14 2024

Corrected link to sequence of indexes. - Serge Batalov, Feb 17 2015

a(10) from Harvey P. Dale, May 10 2016

A139068 Primes of the form k!/9 + 1.

Original entry on oeis.org

4481, 611402462201343216650033936533361654773516861440000000001, 234195255375503079690400057633265510581087082006817356924774723468294901747510352675631491470712754833859385753600000000000000000001

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For numbers k for which (9+k!)/9 is prime see A137390.

Amiram Eldar, Table of n, a(n) for n = 1..9

Cf. A020458, A007749, A082672, A089085, A089130, A117141, A137390, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139066, A137390, A139070, A139071, A139072, A139156.

a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, (n! + 9)/9]], {n, 1, 150}]; a
Select[Range[100]!/9+1,PrimeQ] (* Harvey P. Dale, Aug 17 2017 *)

for(n=6,1e4,if(ispseudoprime(t=n!/9+1),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A139156(A137390(n)). - Amiram Eldar, Oct 14 2024

A139070 Primes of the form (10+k!)/10.

Original entry on oeis.org

13, 73, 3991681, 47900161, 130767436801, 2585201673888497664001, 40329146112660563558400001, 1376375309122634504631597958158090240000001, 11962222086548019456196316149565771506438373376000000001

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For numbers k for which (10+k!)/10 is prime see A139071.

Amiram Eldar, Table of n, a(n) for n = 1..16

Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139066, A139068, A137390, A139071, A139072, A139157.

a = {}; Do[If[PrimeQ[(n! + 10)/10], AppendTo[a, (n! + 10)/10]], {n, 1, 50}]; a
Select[(Range[50]!+10)/10,PrimeQ] (* Harvey P. Dale, Sep 18 2013 *)

for(k=5,1e3,if(ispseudoprime(t=k!/10+1),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A139157(A139071(n)). - Amiram Eldar, Oct 14 2024

A139075 Primes p arising in A139074.

Original entry on oeis.org

3, 2, 3, 31, 1009, 2, 5702401, 631

Offset: 1

Artur Jasinski, Apr 08 2008, Apr 21 2008

nonn

a(23) = (23+1579!)/23. - Andrew V. Sutherland, Apr 11 2008.
Smallest mother factorial prime p of order n, i.e. smallest prime of the form (p!+n)/n where p is prime.
For smallest daughter factorial prime p of order n see A139074.
For smallest father factorial prime p of order n see A139207.
For smallest son factorial prime p of order n see A139206.
a(9)=26737!/9+1 is a 106758 digit (probable) prime.  Easily calculated but too large to enter here a(10)=13, a(11)=566092801, a(12)=11. [Robert Price, Jan 19 2011]

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027.

a = {}; Do[k = 1; While[ ! PrimeQ[(Prime[k]! + n)/n], k++ ]; AppendTo[a, Prime[(Prime[k]! + n)/n]], {n, 1, 8}]; a

A139061 Numbers n for which (4+n!)/4 is prime.

Original entry on oeis.org

4, 5, 6, 13, 21, 25, 32, 40, 61, 97, 147, 324, 325, 348, 369, 1290, 1342, 3167, 6612, 8176, 10990

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For primes of the form (4+k!)/4, see A139060.

a(22) > 25000. - Robert Price, Jan 10 2017

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072.

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! + 4)/4], AppendTo[a, n]], {n, 1, 500}]; a
Select[Range[500],PrimeQ[(4+#!)/4]&]  (* Harvey P. Dale, Mar 24 2011 *)

for(n=4,1e3,if(ispseudoprime(n!/4+1),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

More terms from Serge Batalov, Feb 18 2015

a(19) - a(21) from Robert Price, Jan 10 2017

A139063 Numbers k for which (6+k!)/6 is prime.

Original entry on oeis.org

3, 4, 10, 11, 13, 14, 17, 21, 82, 115, 165, 167, 173, 174, 208, 225, 380, 655, 1187, 2000, 2568, 3010, 4542, 8750, 12257, 12601, 24083

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For primes of the form (6+k!)/6, see A139062.

a(28) > 25000. - Robert Price, Nov 20 2016

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458.

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! + 6)/6], AppendTo[a, n]], {n, 1, 500}]; a

for(k=3,1e3,if(ispseudoprime(k!/6+1),print1(k", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(18) and a(19) from Robert Israel, May 19 2014
More terms from Serge Batalov, Feb 18 2015
a(24)-a(27) from Robert Price, Nov 20 2016

A139065 Numbers k for which (7+k!)/7 is prime.

Original entry on oeis.org

11, 15, 16, 25, 35, 59, 64, 68, 82, 121, 149, 238, 584, 912, 3349, 4111, 4324, 15314, 19944, 20658, 22740, 23364

Offset: 1

Artur Jasinski, Apr 07 2008

For primes of the form (7+k!)/7, see A139064.

a(23) > 25000. - Robert Price, Nov 20 2016

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458.

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! + 7)/7], AppendTo[a, n]], {n, 1, 500}]; a
Select[Range[500],PrimeQ[(7+#!)/7]&] (* Harvey P. Dale, Sep 01 2014 *)

for(k=7,1e3,if(ispseudoprime(k!/7+1),print1(k", "))) \\ Charles R Greathouse IV, Jul 15 2011

More terms from Serge Batalov, Feb 18 2015

a(18)-a(22) from Robert Price, Nov 20 2016

A139157 a(n) = (n!+10)/10.

Original entry on oeis.org

13, 73, 505, 4033, 36289, 362881, 3991681, 47900161, 622702081, 8717829121, 130767436801, 2092278988801, 35568742809601, 640237370572801, 12164510040883201, 243290200817664001, 5109094217170944001

Offset: 5

Artur Jasinski, Apr 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 5..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139156, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 10)/10, {n, 5, 30}]
```

for(n=5, 20, a=(n!+10)/10; print1(a, ", ")) \\ Felix Fröhlich, Jul 07 2014

Name corrected by Amiram Eldar, Oct 14 2024

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A007749 Numbers k such that k!! - 1 is prime.

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A139056 Numbers k for which (k!-3)/3 is prime.

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A139066 Primes of the form (8+k!)/8.

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A139068 Primes of the form k!/9 + 1.

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A139070 Primes of the form (10+k!)/10.

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A139075 Primes p arising in A139074.

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A139061 Numbers n for which (4+n!)/4 is prime.

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A139063 Numbers k for which (6+k!)/6 is prime.

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