A007749 -id:A007749 - cp's OEIS frontend

A139074 a(n) = smallest prime p such that p!/n + 1 is prime, or 0 if no such prime exists.

2, 2, 3, 5, 7, 3, 11, 7, 26737, 5, 13, 5

Offset: 1

Artur Jasinski, Apr 08 2008, Apr 21 2008

For the corresponding primes p see A139075.
a(9)>5000, a(13)>5000, a(22)>5000, a(23) = 1579. - Andrew V. Sutherland, Apr 21 2008, Apr 22 2008
a(10)=5,   a(11)=13, a(12)=5
a(14)=17,  a(15)=7,  a(16)=13,  a(17)=43, a(18)=7,
a(19)=31,  a(20)=5,  a(21)=7
a(24)=7,   a(25)=47, a(26)=17,  a(27)=17, a(28)=7,
a(29)=241, a(30)=5,  a(31)=61,  a(32)=11, a(33)=17,
a(34)=17,  a(35)=29, a(36)=11,  a(37)=61, a(38)=103,
a(39)=89,  a(40)=7,  a(41)=131, a(42)=11, a(43)=71,
a(44)=13,  a(45)=7,  a(46)=43,  a(47)=73, a(48)=67,
a(49)=347, a(50)=31, a(51)=19,  a(52)=17, a(53)=347,
a(54)=11,  a(55)=13, a(56)=13,  a(57)=31, a(58)=73,
a(59)=641, a(60)=5
a(23) = 1579. - Andrew V. Sutherland, Apr 11 2008.
Smallest daughter factorial prime p of order n, i.e. smallest prime of the form (p!+n)/n where p is prime.
For smallest mother factorial prime p of order n see A139075
For smallest father factorial prime p of order n see A139207
For smallest son factorial prime p of order n see A139206
Summary added by Robert Price, Nov 25 2010:
a(1:20)=2,2,3,5,7,3,11,7,26737,5,13,5,>60000,17,7,13,43,7,31,5
a(21:40)=7,>60000,1579,7,47,17,17,7,241,5,61,11,17,17,29,11,61,103,89,7
a(41:60)=131,11,71,13,7,43,73,67,347,31,19,17,347,11,13,13,31,73,641,5
a(61:80)=89,31,13,13,17,11,71,19,131,7,151,7,>10000,641,73,43,17,331,113,11
a(81:100)=13,67,>10000,7,1999,89,31,11,>10000,19,19,31,607,71,61,11,761,23,>10000,83

			a(1) = 2 because 2 is the first prime and 2!/1 + 1 = 3 is prime
a(2) = 2 because 2 is the first prime and 2!/2 + 1 = 2 is prime
a(3) = 3 because 3!/3 + 1 = 3 is prime

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[k]], {n, 1, 8}]; a

a(9)-a(12) by Robert Price, Dec 19 2010

A139156 a(n) = (n!+9)/9.

Original entry on oeis.org

81, 561, 4481, 40321, 403201, 4435201, 53222401, 691891201, 9686476801, 145297152001, 2324754432001, 39520825344001, 711374856192001, 13516122267648001, 270322445352960001, 5676771352412160001

Offset: 6

Artur Jasinski, Apr 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 6..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 9)/9, {n, 6, 30}]
```

Name corrected by Amiram Eldar, Oct 14 2024

A262772 Numbers k such that k!! - 32 is prime.

Original entry on oeis.org

7, 45, 67, 145, 411, 825, 1021, 4039, 9069, 9789, 12463, 15137, 26313, 27499

Offset: 1

Robert Price, Sep 30 2015

Corresponding primes are 73, 25373791335626257947657609343, ... .

a(15) > 50000.

Joe McLean, Interesting Sources of Probable Primes

Cf. A006882, A007749, A094144, A123910, A258616.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ n: n in [7..450] | IsPrime(DoubleFactorial(n) -32) ]; // Vincenzo Librandi, Oct 01 2015

Select[Range[0, 50000], If[#!! - 32 > 0, PrimeQ[#!! - 32]] &]

for(n=1, 1e4, if (isprime(prod(k=0, (n-1)\2, n - 2*k ) - 32),print1(n", "))) \\ Altug Alkan, Oct 01 2015

A139092 a(n) = number of distinct prime divisors of (9+prime(n)!)/9.

Original entry on oeis.org

3, 3, 2, 2, 2, 3, 2, 3, 4, 4, 5, 3, 3, 6, 5, 2, 3, 4, 3, 3, 4, 4, 4, 3, 7, 3, 3

Offset: 4

Artur Jasinski, Apr 08 2008

Conjecture: all prime divisors in A139089 are distinct

a(31) >= 4. - Amiram Eldar, Feb 13 2020

FactorDB, Status of a(31) = (9+127!)/9.

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027, A139089-A139092.

a = {}; Do[w = (Prime[n]! + 9)/9; AppendTo[a, w], {n, 4, 16}]; a
PrimeNu[(9+Prime[Range[4,25]]!)/9] (* Harvey P. Dale, Jul 25 2019 *)

a(n) = A001221(A139089(n)). - Amiram Eldar, Feb 13 2020

More terms from Jon E. Schoenfield, Jul 16 2010

a(23)-a(30) using factordb.com from Amiram Eldar, Feb 13 2020

A139149 a(n) = (n!+2)/2.

Original entry on oeis.org

2, 4, 13, 61, 361, 2521, 20161, 181441, 1814401, 19958401, 239500801, 3113510401, 43589145601, 653837184001, 10461394944001, 177843714048001, 3201186852864001, 60822550204416001, 1216451004088320001, 25545471085854720001, 562000363888803840001

Offset: 2

Artur Jasinski, Apr 11 2008

Also the number of (not necessarily maximal) cliques in the (n-1)-(weak) Bruhat graph. - Eric W. Weisstein, Jul 29 2018

			(1!+2)/2 = 3/2 is not an integer.
a(2) = (2!+2)/2 = 2.

Amiram Eldar, Table of n, a(n) for n = 2..450
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Clique.

a(n) = (n!+m)/m: A038507 (m=1), this sequence (m=2), A139150 (m=3), A139151 (m=4), A139152 (m=5), A139153 (m=6), A139154 (m=7), A139155 (m=8), A139156 (m=9), A139157 (m=10).
Offsets for above sequences are Kempner numbers A002034.
For smallest number of the form (m!+n)/n see A139148.
Cf. A007749, A020458, A082672, A089085, A089130, A117141, A137390, A139056-A139066, A139068, A139070-A139075, A139157, A139159-A139162.

Mathematica
```
Table[(n! + 2)/2, {n, 2, 30}]
```

a(n)=n!/2+1 \\ Charles R Greathouse IV, Jun 20 2012

A139150 a(n) = (n!+3)/3.

Original entry on oeis.org

3, 9, 41, 241, 1681, 13441, 120961, 1209601, 13305601, 159667201, 2075673601, 29059430401, 435891456001, 6974263296001, 118562476032001, 2134124568576001, 40548366802944001, 810967336058880001

Offset: 3

Artur Jasinski, Apr 11 2008

nonn

			(2!+3)/3 = 5/3 is not an integer.
a(3) = (3!+3)/3 = 3.

Amiram Eldar, Table of n, a(n) for n = 3..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 3)/3, {n, 3, 30}]
```

Name corrected by Amiram Eldar, Oct 13 2024

A139152 a(n) = (n!+5)/5.

Original entry on oeis.org

25, 145, 1009, 8065, 72577, 725761, 7983361, 95800321, 1245404161, 17435658241, 261534873601, 4184557977601, 71137485619201, 1280474741145601, 24329020081766401, 486580401635328001, 10218188434341888001

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-A139157, A139159, A139160-A139162.

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

Name corrected by Amiram Eldar, Oct 14 2024

A139153 a(n) = (n!+6)/6.

Original entry on oeis.org

2, 5, 21, 121, 841, 6721, 60481, 604801, 6652801, 79833601, 1037836801, 14529715201, 217945728001, 3487131648001, 59281238016001, 1067062284288001, 20274183401472001, 405483668029440001, 8515157028618240001

Offset: 3

Artur Jasinski, Apr 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 3..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 6)/6, {n, 3, 30}]
```

Name corrected by Amiram Eldar, Oct 14 2024

A139154 a(n) = (n!+7)/7.

Original entry on oeis.org

721, 5761, 51841, 518401, 5702401, 68428801, 889574401, 12454041601, 186810624001, 2988969984001, 50812489728001, 914624815104001, 17377871486976001, 347557429739520001, 7298706024529920001

Offset: 7

Artur Jasinski, Apr 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 7..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 7)/7, {n, 7, 30}]
```

Name corrected by Amiram Eldar, Oct 14 2024

A139155 a(n) = (n!+8)/8.

Original entry on oeis.org

4, 16, 91, 631, 5041, 45361, 453601, 4989601, 59875201, 778377601, 10897286401, 163459296001, 2615348736001, 44460928512001, 800296713216001, 15205637551104001, 304112751022080001, 6386367771463680001

Offset: 4

Artur Jasinski, Apr 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 4..450

Cf. A007749, A020458, A082672, A089085, A089130, A117141, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160, A139161, A139162.

Mathematica
```
Table[(n! + 8)/8, {n, 4, 30}]
```

Name corrected by Amiram Eldar, Oct 14 2024

cp's OEIS Frontend

A139074 a(n) = smallest prime p such that p!/n + 1 is prime, or 0 if no such prime exists.

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A139156 a(n) = (n!+9)/9.

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A262772 Numbers k such that k!! - 32 is prime.

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A139092 a(n) = number of distinct prime divisors of (9+prime(n)!)/9.

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A139149 a(n) = (n!+2)/2.

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A139150 a(n) = (n!+3)/3.

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A139152 a(n) = (n!+5)/5.

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Extensions

A139153 a(n) = (n!+6)/6.

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