A137390 -id:A137390 - cp's OEIS frontend

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

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

A139163 a(n) = (prime(n)!+5)/5.

Original entry on oeis.org

25, 1009, 7983361, 1245404161, 71137485619201, 24329020081766401, 5170403347776995328001, 1768352398747940390908723200001, 1644567730835584563545112576000001

Offset: 3

Artur Jasinski, Apr 11 2008

nonn

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160-A139166, A139089, A139168-A139170.

A139163:=n->(ithprime(n)! + 5)/5; seq(A139163(n), n=3..15); # Wesley Ivan Hurt, Jan 23 2014

Mathematica
```
Table[(Prime[n]! + 5)/5, {n, 3, 30}]
```

a(n) = (prime(n)!+5)/5; \\ Michel Marcus, Jan 23 2014

A139169 a(n)=smallest k >= 1 such that n divides prime(k)!.

Original entry on oeis.org

1, 1, 2, 3, 3, 2, 4, 3, 4, 3, 5, 3, 6, 4, 3, 4, 7, 4, 8, 3, 4, 5, 9, 3, 5, 6, 5, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 3, 13, 4, 14, 5, 4, 9, 15, 4, 7, 5, 7, 6, 16, 5, 5, 4, 8, 10, 17, 3, 18, 11, 4, 5, 6, 5, 19, 7, 9, 4, 20, 4, 21, 12, 5, 8, 5, 6, 22, 4, 5, 13, 23, 4, 7, 14, 10, 5, 24, 4, 6, 9, 11, 15

Offset: 1

Artur Jasinski, Apr 11 2008

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Indices of primes in A139171.

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160-A139166, A139089, A139168-A139170.

f:= proc(n) local F,m,Q,E,p;
  F:= ifactors(n)[2];
  m:= nops(F);
  Q:= map(t -> t[1],F);
  E:= map(t -> t[2],F);
  p:= max(Q)-1;
  do
    p:= nextprime(p);
    if andmap(i -> add(floor(p/Q[i]^j),j=1..floor(log[Q[i]](p))) >= E[i], [$1..m]) then return p fi;
  od
end proc:
f(1):= 2:
map(numtheory:-pi @ f, [$1..100]); # Robert Israel, Mar 07 2018

a = {}; Do[m = 1; While[ ! IntegerQ[Prime[m]!/n], m++ ]; AppendTo[a, m], {n, 1, 100}]; a

a(n) = forprime(p=2,, if (!(p! % n), return (primepi(p)))); \\ Michel Marcus, Mar 08 2018

A139171 a(n) = smallest prime number p such that p!/n is an integer.

Original entry on oeis.org

2, 2, 3, 5, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 5, 7, 17, 7, 19, 5, 7, 11, 23, 5, 11, 13, 11, 7, 29, 5, 31, 11, 11, 17, 7, 7, 37, 19, 13, 5, 41, 7, 43, 11, 7, 23, 47, 7, 17, 11, 17, 13, 53, 11, 11, 7, 19, 29, 59, 5, 61, 31, 7, 11, 13, 11, 67, 17, 23, 7, 71, 7, 73, 37, 11, 19, 11, 13, 79, 7, 11

Offset: 1

Artur Jasinski, Apr 11 2008

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Legendre's formula

Prime equivalent of Kempner numbers A002034.
For quotients p!/n see A139170.
For indices of primes in this sequence see A139169.
Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160-A139166, A139089, A139168-A139170.

f:= proc(n) local F,m,Q,E,p;
  F:= ifactors(n)[2];
  m:= nops(F);
  Q:= map(t -> t[1],F);
  E:= map(t -> t[2],F);
  p:= max(Q)-1;
  do
    p:= nextprime(p);
    if andmap(i -> add(floor(p/Q[i]^j),j=1..floor(log[Q[i]](p))) >= E[i], [$1..m]) then return p fi;
  od
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Mar 07 2018

a = {}; Do[m = 1; While[ ! IntegerQ[Prime[m]!/n], m++ ]; AppendTo[a, Prime[m]], {n, 1, 100}]; a

a(n) = forprime(p=2,, if (!(p! % n), return (p))); \\ Michel Marcus, Mar 08 2018

A139200 Numbers k such that (k!-5)/5 is prime.

Original entry on oeis.org

5, 11, 12, 16, 36, 41, 42, 47, 127, 136, 356, 829, 1863, 2065, 2702, 4509, 7498

Offset: 1

Artur Jasinski, Apr 11 2008

a(16) > 3000. - Ray G. Opao, Oct 05 2008

a(18) > 25000. - Robert Price, Nov 20 2016

Cf. A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

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).

[n: n in [5..500] | IsPrime((Factorial(n)-5) div 5)]; // Vincenzo Librandi, Nov 21 2016

a = {}; Do[If[PrimeQ[(n! - 5)/5], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski *)

a(13)-a(15) from Ray G. Opao, Oct 05 2008
a(16) from Serge Batalov, Feb 18 2015
a(17) from Robert Price, Nov 20 2016

cp's OEIS Frontend

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

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Mathematica

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

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Mathematica

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

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Mathematica

Extensions

A139163 a(n) = (prime(n)!+5)/5.

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Maple

Mathematica

PARI

A139169 a(n)=smallest k >= 1 such that n divides prime(k)!.

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Maple

Mathematica

PARI

A139171 a(n) = smallest prime number p such that p!/n is an integer.

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