A002981 -id:A002981 - cp's OEIS frontend

A139203 Numbers k such that (k!-8)/8 is prime.

4, 6, 8, 10, 11, 16, 19, 47, 66, 183, 376, 507, 1081, 1204, 12111, 23181

Offset: 1

Artur Jasinski, Apr 11 2008

a(17) > 25000. - Robert Price, Oct 08 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((1/8)*factorial(n)-1)=true then n else end if end proc: seq(a(n),n=4..550); # Emeric Deutsch, May 07 2008

a = {}; Do[If[PrimeQ[(n! - 8)/8], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a

2 more terms from Emeric Deutsch, May 07 2008
More terms from Serge Batalov, Feb 18 2015
a(15)-a(16) from Robert Price, Oct 08 2016

A139204 Numbers k such that (k!-9)/9 is prime.

Original entry on oeis.org

6, 15, 17, 18, 21, 27, 29, 30, 37, 47, 50, 64, 125, 251, 602, 611, 1184, 1468, 5570, 10679, 15798, 21237

Offset: 1

Artur Jasinski, Apr 11 2008

a(20) > 10000. The PFGW program has been used to certify all the terms up to a(19), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Mar 28 2014

a(23) > 25000. - Robert Price, Mar 29 2017

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! - 9)/9], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a

for(n=1,1000,if(floor(n!/9-1)==n!/9-1,if(ispseudoprime(n!/9-1),print(n)))) \\ Derek Orr, Mar 28 2014

a(14)-a(16) from Derek Orr, Mar 28 2014
a(17)-a(19) from Giovanni Resta, Mar 28 2014
a(20)-a(22) from Robert Price, Mar 29 2017

A151894 Numbers k such that k! + second prime after k! is prime.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 11, 17, 42, 66, 76, 93, 139, 157, 226, 290, 415, 522, 774, 794, 1947

Offset: 1

Artur Jasinski, Apr 12 2008

Because numbers of the form (k! + prime) are divisible by all primes <= k that means that the first prime number can have the form k! + next prime after k! and no primes of the form k! + m for m > 1 and m < next prime after k!.

Cf. A002981, A151892, A151893, A151903.

a = {}; Do[If[PrimeQ[n! + NextPrime[n!,2]], AppendTo[a, n]], {n, 200}]; a

is(k) = {my(f = k!); isprime(f + nextprime(nextprime(f + 1) + 1));} \\ Amiram Eldar, Oct 23 2024

a(15)-a(20) from Amiram Eldar, Oct 23 2024

a(21) from Michael S. Branicky, Oct 24 2024

A163077 Numbers k such that k$ + 1 is prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 14, 15, 24, 27, 31, 38, 44, 45, 49, 67, 76, 92, 99, 119, 124, 133, 136, 139, 144, 168, 171, 185, 265, 291, 332, 368, 428, 501, 631, 680, 689, 696, 765, 789, 890, 1034, 1233, 1384, 1517, 1615, 1634, 1809, 2632, 2762, 3925, 4419, 5108, 5426

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			0$ + 1 = 1 + 1 = 2 is prime, so 0 is in the sequence.

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

Cf. A002981, A163078, A163079, A163080.

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

fQ[n_] := PrimeQ[1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range[0, 8660], fQ] (* Robert G. Wilson v, Aug 09 2010 *)

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

a(45)-a(56) from Robert G. Wilson v, Aug 09 2010

A231901 Least k > n such that k!/n! + 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 11, 9, 11, 10, 20, 12, 15, 15, 16, 16, 18, 18, 23, 21, 22, 22, 40, 25, 27, 31, 28, 28, 37, 30, 42, 38, 34, 36, 42, 36, 110, 39, 43, 40, 42, 42, 56, 46, 50, 46, 55, 65, 51, 51, 53, 52, 55, 55, 73, 58, 58, 58, 60, 60, 63, 63, 177, 68, 70, 66, 82, 72

Offset: 0

Alex Ratushnyak, Nov 15 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002981, A035093, A057217, A075757, A075758, A231549.

Table[k = n + 1; While[! PrimeQ[k!/n! + 1], k++]; k, {n, 100}] (* T. D. Noe, Nov 18 2013 *)

a(n) = {my(m = n+1); while(! isprime(m!/n! +1), m++); m;} \\ Michel Marcus, Mar 07 2014; corrected Jun 13 2022

A242487 Numbers k such that (2*k)! + k! + 1 is prime.

Original entry on oeis.org

0, 3, 8, 13, 19, 423, 585, 2746, 2855

Offset: 1

Seiichi Manyama, Mar 22 2018

a(10) > 10000. - Michael S. Branicky, May 03 2025

			0! + 0! + 1 =   3 is prime.
6! + 3! + 1 = 727 is prime.

A300947 gives the primes.

Cf. A002981, A237443.

select(k->isprime(factorial(2*k)+factorial(k)+1),[$0..600]); # Muniru A Asiru, May 27 2018

Flatten[{0, Select[Range[100], PrimeQ[(2*#)! + #! + 1] &]}] (* Vaclav Kotesovec, Mar 25 2018 *)

isok(k) = ispseudoprime((2*k)!+k!+1); \\ Altug Alkan, Mar 22 2018

a(8)-a(9) from Michael S. Branicky, Apr 16 2023

A340013 The prime gap, divided by two, which surrounds n!.

Original entry on oeis.org

1, 3, 7, 4, 6, 27, 15, 11, 7, 15, 45, 10, 45, 38, 45, 39, 95, 30, 31, 52, 93, 102, 95, 48, 22, 84, 127, 54, 94, 40, 19, 145, 87, 129, 49, 22, 85, 68, 66, 88, 90, 78, 146, 95, 156, 78, 71, 79, 225, 60, 65, 175, 66, 305, 192, 196, 215, 205, 420, 101, 186, 213, 160

Offset: 3

Robert G. Wilson v, Jan 09 2021

nonn

A theorem states that between (n+1)! + 2 and (n+1)! + (n+1) inclusive, there are n consecutive composite integers, namely 2, 3, 4, ..., n, n+1.

Records: 1, 3, 7, 27, 45, 95, 102, 127, 145, 146, 156, 225, 305, 420, 804, 844, 1173, 1671, 1725, 1827, 2570, 2930, 3318, 5142, 5946, 6837, 7007, 8208, 10221, ..., .

			For a(1), there are no positive primes which surround 1!. Therefore a(1) is undefined.
For a(2), there are two contiguous primes {2, 3} with 2 being 2!. The prime gap is 1. However, the two primes do not surround 2!, so a(2) is undefined.
For a(3), the following set of numbers, {5, 6, 7}, with 3! being in the middle. The prime gap is 2; therefore, a(3) = 1.
For a(4), the following set of numbers, {23, 24, 25, 26, 27, 28, 29} with 4! in between the two primes 23 & 29. The prime gap is 6, so a(4) = 3.

Robert Israel, Table of n, a(n) for n = 3..607

Cf. A000142, A002981, A006990, A033932, A033933, A037151, A054588, A058054, A073308.

a:= n-> (f-> (nextprime(f-1)-prevprime(f+1))/2)(n!):
seq(a(n), n=3..70);  # Alois P. Heinz, Jan 09 2021

a[n_] := (NextPrime[n!, 1] - NextPrime[n!, -1])/2; Array[a, 70, 3]

a(n) = (nextprime(n!+1) - precprime(n!-1))/2; \\ Michel Marcus, Jan 11 2021

from sympy import factorial, nextprime, prevprime
def A340013(n):
    f = factorial(n)
    return (nextprime(f)-prevprime(f))//2 # Chai Wah Wu, Jan 23 2021

a(n) = (A037151(n) - A006990(n))/2 = (A033932(n) + A033933(n))/2.

a(n) = A054588(n)/2 = A058054(n)/2. - Alois P. Heinz, Jan 09 2021

A064295 Numbers k such that every (prime) factor of k!+1 is one more than a multiple of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 25, 27, 28, 37, 41, 46, 73, 77, 116

Offset: 1

Don Reble, Sep 25 2001

a(20) >= 140. - Max Alekseyev, Feb 19 2024

			46 is in the list because 46!+1 = 47 * 268662306503771535067 * 435777793891607546778854755077304349 and all of those factors are = 1 mod 46.

H. Mishima, Factors of N!+1

Complement of A064164, superset of A002981.

a(19) from Amiram Eldar, Oct 03 2019

A084727 Primes arising in A084726.

Original entry on oeis.org

2, 3, 7, 281, 76561, 576577, 17873857, 643458817, 337767408001, 21617114112001, 39916801, 119715577952256001, 1980990543353657472001, 26582634158080001, 3577861898239093446857008573440001, 711975497511453268455460274177

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

nonn

Smallest prime of the form: 1 + product of n terms of an arithmetic progression with first term 1.
Conjecture: All terms exist.
If n! + 1 is prime (A002981) then a(n) = A088332(n). - Hugo Pfoertner, Nov 18 2004

			a(1) = 2 = 1 + 1;
a(4) = 281 = 1*4*7*10 + 1 (1*2*3*4 + 1 = 25 is composite);
a(5) = 76561 = 1*8*15*22*29 + 1.

Nathaniel Johnston, Table of n, a(n) for n = 1..100

Cf. A002981, A084726, A088332.

A084727 := proc(n) local k,p: for k from 1 do p:=1+mul(1+j*k,j=0..n-1): if(isprime(p))then return p: fi: od: end: seq(A084727(n),n=1..16); # Nathaniel Johnston, Jun 26 2011

np[n_]:=Module[{k=1},While[!PrimeQ[Times@@NestList[k+#&,1,n-1]+1],k++];Times@@NestList[k+#&,1,n-1]+1]; Array[np,20] (* Harvey P. Dale, Aug 05 2021 *)

a(n) = 1 + Product_{i = 0..n-1} (1 + i*A084726(n)). - David Wasserman, Jan 03 2005

More terms from David Wasserman, Jan 03 2005

A093621 Smallest k>0 such that n!/k!+1 is prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 2, 2, 4, 3, 1, 5, 2, 3, 4, 2, 3, 5, 19, 10, 3, 5, 4, 5, 7, 5, 1, 6, 21, 2, 9, 15, 15, 13, 10, 27, 1, 4, 37, 14, 1, 4, 2, 34, 5, 8, 18, 24, 2, 13, 5, 11, 35, 48, 11, 7, 48, 27, 21, 30, 5, 43, 7, 4, 46, 13, 24, 16, 60, 12, 34, 5, 1, 38, 14, 28, 1, 10, 24, 50, 5, 3, 42, 40, 28

Offset: 0

Hugo Pfoertner, Apr 06 2004

nonn

The results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

			a(39) =37 because 39!/k!+1 is composite for all k=1..36; 39!/37!+1=39*38+1=1483 is prime.

Cf. A093437 largest prime of the form n!/k!+1, A002981 n!+1 is prime, A093623 Smallest k>0 such that n!/k!-1 is prime.

a(A002981(n)) = 1.

cp's OEIS Frontend

A139203 Numbers k such that (k!-8)/8 is prime.

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A139204 Numbers k such that (k!-9)/9 is prime.

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A151894 Numbers k such that k! + second prime after k! is prime.

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A163077 Numbers k such that k$ + 1 is prime. Here '$' denotes the swinging factorial function (A056040).

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A231901 Least k > n such that k!/n! + 1 is a prime, or 0 if no such k exists.

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A242487 Numbers k such that (2*k)! + k! + 1 is prime.

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A340013 The prime gap, divided by two, which surrounds n!.

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A064295 Numbers k such that every (prime) factor of k!+1 is one more than a multiple of k.

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