A117141 -id:A117141 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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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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

Extensions

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

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Programs

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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