A002981 -id:A002981 - cp's OEIS frontend

A151903 a(n) = smallest number k such that n! + k-th prime after n is prime.

1, 1, 1, 1, 1, 1, 1, 5, 3, 1, 2, 5, 13, 2, 8, 3, 4, 5, 16, 2, 3, 3, 16, 23, 4, 8, 6, 10, 38, 18, 20, 11, 1, 14, 7, 21, 52, 2, 13, 4, 5, 12, 6, 1, 38, 12, 36, 1, 8, 3, 43, 1, 4, 32, 4, 19, 12, 45, 45, 41, 118, 14, 40, 1, 26, 43, 2, 4, 13, 15, 128, 6, 1, 20, 29, 9, 14, 9, 36, 6, 104, 9, 14, 26, 9

Offset: 1

Artur Jasinski, Apr 12 2008

nonn

Because numbers of the form : (n! + prime) are divisible by all primes <= n that mean that first prime number can have form n! + k-th prime after n and no primes of the form n! + k for k > 1 and k < next prime after n

Amiram Eldar, Table of n, a(n) for n = 1..402

Cf. A002981, A151892, A151894, A151893, A139213, A139214.

a = {}; Do[k = 1; While[ ! PrimeQ[n! + NextPrime[n, k]], k++ ]; AppendTo[a, k], {n, 1, 200}]; a

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 5, 31, 67, 139, 631, 9743, 16253, 17977, 27901, 37589

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163077.

			5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A062363, A093804, A163077.

Cf. A056040. - Robert G. Wilson v, Aug 09 2010

a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* Robert G. Wilson v, Aug 08 2010 *)

is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(8)-a(12) from Robert G. Wilson v, Aug 08 2010

A051855 Numbers n such that (n!)^4+1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 13, 112, 328, 11123

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

nonn

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [?Broken link]

Cf. A002981, A002982, A046029.

[n: n in [1..300] | IsPrime(Factorial(n)^4+1)]; // Vincenzo Librandi, Aug 15 2013

Select[Range[0, 350], PrimeQ[(#!)^4 + 1]&] (* Vincenzo Librandi, Aug 15 2013 *)

isok(n) = isprime(n!^4 + 1); \\ Michel Marcus, Aug 15 2013

a(9) from Robert Price, Jul 24 2014

Prepended a(1)=0, Robert Price, Sep 01 2014

A062701 Index of factorial primes of the form k! + 1.

Original entry on oeis.org

1, 2, 4, 2428957

Offset: 1

Labos Elemer, Jul 11 2001

			The exact subscript of the 5th prime [1 + 27! = 10888869450418352160768000001] is not yet available.

Cf. A000720, A002981, A002982, A055490, A088332.

a(n) = PrimePi(A002981(n)!+1).

Offset 1 from Michel Marcus, Aug 29 2019

A082952 Smaller of the two factors of the n-th semiprime number of the form m!+1.

Original entry on oeis.org

5, 11, 7, 71, 61, 11, 83, 23, 71, 20639383, 811, 401, 1697, 29, 67411, 14029308060317546154181, 12893, 12318573951317236818169524329, 79, 16567, 6653, 293, 229758023927584562777368125832724248866067995638905559798117

Offset: 1

Hugo Pfoertner, May 26 2003

nonn

			a(3)=7 because A078778(3)!+1=6!+1=721=7*103

Paul Leyland, Tables of factors of N!+1 and N!-1.
Hisanori Mishima, n! + 1 (n = 1 to 100).

Cf. A046315, A078778, A002981, A080802.

FactorInteger[#][[1,1]]&/@Select[Range[50]!+1,PrimeOmega[#]==2&] (* The program generates the first 17 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run.*) (* Harvey P. Dale, Dec 11 2023 *)

Numbers p such that p*q=A078778(n)!+1, p, q prime, p

A093437 a(n) = largest prime of the form n!/k! + 1.

Original entry on oeis.org

2, 2, 3, 7, 13, 61, 31, 2521, 20161, 15121, 604801, 39916801, 3991681, 3113510401, 14529715201, 54486432001, 10461394944001, 59281238016001, 53353114214401, 2, 670442572801, 8515157028618240001, 9366672731480064001

Offset: 0

Amarnath Murthy, Apr 01 2004

nonn

Is 19 the largest n such that a(n) = 2? There are none for 19 < n <= 600. - Robert Israel, Jan 16 2017

			a(7) = 2521 because 7!/2! + 1 = 2521 is prime, whereas 7!/1! + 1 = 5041 = 71^2 is composite;
a(19) = 2 because the only prime of the form 19!/k! + 1 is 19!/19! + 1 = 2.

Robert Israel, Table of n, a(n) for n = 0..466

Cf. A093621 (smallest k > 0 such that n!/k! + 1 is prime), A002981 (n! + 1 is prime), A088332 (primes of form n! + 1).

f:= proc(n) local k,x;
  x:= n!;
  for k from 2 do
    if isprime(x+1) then return x+1 fi;
    x:= x/k;
  od
end proc:
map(f, [$0..40]); # Robert Israel, Jan 16 2017

a[n_] := Module[{k, x}, x = n!; For[k = 2, True, k++, If[PrimeQ[x+1], Return[x+1]]; x = x/k]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2023, after Robert Israel *)

Corrected and extended by Hugo Pfoertner, Apr 06 2004

A103319 Primes of the form p! + 1 where p is prime.

Original entry on oeis.org

3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000001

Offset: 1

Jonathan Sondow, Jan 31 2005

The values of p are 2, 3, 11, 37, 41, 73 which is A093804 (with a different definition). Subsequence of A088332 (primes of the form n! + 1).

			2 and 2! + 1 = 3 are prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A093804, A002981, A088332, A103318.

Select[Table[p!+1,{p,Prime[Range[30]]}],PrimeQ] (* Harvey P. Dale, Nov 28 2019 *)

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

A139201 Numbers k such that (k!-6)/6 is prime.

Original entry on oeis.org

4, 5, 7, 8, 11, 14, 16, 17, 18, 20, 43, 50, 55, 59, 171, 461, 859, 2830, 3818, 5421, 5593, 10118, 10880, 24350

Offset: 1

Artur Jasinski, Apr 11 2008

nonn

a(25) > 25000. - Robert Price, Dec 15 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).

a:=proc(n) if isprime((1/6)*factorial(n)-1)=true then n else end if end proc: seq(a(n),n=4..500); # Emeric Deutsch, Apr 29 2008

a = {}; Do[If[PrimeQ[(n! - 6)/6], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski *)

2 more terms from Emeric Deutsch, Apr 29 2008
More terms from Serge Batalov, Feb 18 2015
a(22)-a(24) from Robert Price, Dec 15 2016

A139202 Numbers k such that (k!-7)/7 is prime.

Original entry on oeis.org

7, 9, 20, 23, 46, 54, 57, 71, 85, 387, 396, 606, 1121, 2484, 6786, 9321, 11881, 18372

Offset: 1

Artur Jasinski, Apr 11 2008

a(19) > 25000. - Robert Price, Nov 05 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).

a = {}; Do[If[PrimeQ[(n! - 7)/7], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (*Artur Jasinski*)

More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 14 2008
a(13)-a(14) PRPs from Sean A. Irvine, Aug 05 2010
a(15)-a(18) PRP from Robert Price, Nov 05 2016

Previous Showing 61-70 of 111 results. Next

cp's OEIS Frontend

A151903 a(n) = smallest number k such that n! + k-th prime after n is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A051855 Numbers n such that (n!)^4+1 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A062701 Index of factorial primes of the form k! + 1.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A082952 Smaller of the two factors of the n-th semiprime number of the form m!+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A093437 a(n) = largest prime of the form n!/k! + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A103319 Primes of the form p! + 1 where p is prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments