A073308 -id:A073308 - cp's OEIS frontend

A090786 Least nonnegative integer k such that n! + n + k + 1 is prime.

0, 0, 0, 1, 0, 1, 0, 3, 14, 7, 0, 5, 16, 53, 4, 27, 6, 13, 18, 69, 8, 9, 8, 73, 106, 15, 32, 19, 38, 193, 76, 95, 46, 3, 62, 25, 94, 273, 4, 57, 12, 19, 54, 27, 2, 193, 54, 185, 4, 33, 10, 219, 0, 17, 168, 15, 92, 49, 224, 233, 210, 707, 68, 207, 2, 127, 216, 5, 14, 61, 68, 785

Offset: 0

Frederick Magata (frederick.magata(AT)uni-muenster.de), Feb 09 2004

nonn

The (n-1) consecutive numbers n!+2, ..., n!+n (for n >= 2) are not prime. This fact implies that there are arbitrarily large gaps in the distribution of the prime numbers. However, n!+n+1 need not be a prime number. Now a(n) measures, when the next prime number after n!+n appears. Thus a(n)=0 means that n!+n+1 is prime and so on. Obviously, a(n) is parity conserving for n >= 2. I.e., if n >= 2 then 2 divides n iff 2 divides a(n).

Conjectures: By definition a(n)+n!+1 is prime, but is a(n)+n+1=A037153(n) also a prime number for all n > 2? Is the growth of b(n) := Sum_{k=0..n} a(k) quadratic, that is, is b(n)=O(n^2)?

			a(5)=1 because 5!+5+1+1=127 is prime and 126 is not.
a(7)=3 because 7!+7+3+1=5051 is prime and 5048, 5049 and 5050 are not prime.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A005095, A037153, A073308.

a := proc(n) option remember;local r,m,k: r := n!+n: k := 1: m := r+1: while (not isprime(m)) do k := k+1: while (not igcd(k,n)=1) do k := k+1: od: m := r+k: od: k-1; end;
# alternatively:
a := proc(n) option remember; nextprime(n!+n)-n!-n-1; end;

lnik[n_]:=Module[{c=n!+n+1},If[PrimeQ[c],0,NextPrime[c]-c]]; Array[ lnik, 80,0] (* Harvey P. Dale, Apr 08 2019 *)

a(n) = apply(x->(nextprime(x)-x), n!+n+1); \\ Michel Marcus, Mar 21 2018

A073443 Numbers k such that k! - k - 1 is prime.

Original entry on oeis.org

3, 4, 10, 12, 346

Offset: 1

Rick L. Shepherd, Jul 31 2002

Clearly n <> 2 (mod 3). For n>3, n!-n, n!-n+1, ..., n!-3, n!-2 is a sequence of n-1 consecutive composite numbers. Additional terms are greater than 2000.
a(5) > 7500. - Michael S. Branicky, Mar 04 2021
a(5) > 24000. - Michael S. Branicky, Nov 13 2024

Cf. A073444 (corresponding primes), A002982 (n!-1 is prime), A073308 (n!+n+1 is prime).

Select[Range[3, 346], PrimeQ[#! - # - 1] &] (* Arkadiusz Wesolowski, Jan 04 2012 *)

for(n=3,2000,if(isprime(n!-n-1),print1(n,",")))

from math import factorial
from sympy import isprime
def ok(n): return isprime(factorial(n) - n - 1)
print([m for m in range(3, 500) if ok(m)]) # Michael S. Branicky, Mar 04 2021

Offset corrected by Arkadiusz Wesolowski, Jan 04 2012

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

A073309 Primes of the form k! + k + 1.

Original entry on oeis.org

2, 3, 5, 29, 727, 3628811, 80658175170943878571660636856403766975289505440883277824000000000053

Offset: 1

Rick L. Shepherd, Jul 24 2002

nonn

a(6) = 3628811 and a(7), a 68-digit number, have been certified prime with Primo.

			a(4) = 6! + 6 + 1 = 727, a prime, so 727 is in this sequence (6 = A073308(4)).

Cf. A073308 (corresponding n), A100858.

f[n_]:=n!+n+1; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,0,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 02 2009 *)

for(n=0,1960,p=n!+n+1; if(isprime(p),print1(p,",")))

a(n) = A073308(n)! + A073308(n) + 1.

A092791 Numbers k such that (k-1)! + k is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 53, 6823, 30839

Offset: 1

Cino Hilliard, Apr 15 2004

nonn

All terms beyond the first must be prime. [Charles R Greathouse IV, Apr 19 2013]

a(10) > 37000. - Giovanni Resta, May 04 2013

Cf. A073308.

Select[Range[80], PrimeQ[(# - 1)! + #] &] (* Jayanta Basu, Apr 19 2013 *)

pm1factpp(n) = { forprime(p=1,n, c=0; y=(p-1)!+p;if(isprime(y),print1(p",")) ) }

a(n) = A073308(n) + 1. - Michael S. Branicky, May 06 2025

a(8)-a(9) from Giovanni Resta, May 04 2013

A192367 Numbers k such that k! + 2*k + 1 is prime.

Original entry on oeis.org

0, 2, 3, 5, 6, 18, 44, 113, 2615, 16914

Offset: 1

Julio Cesar Hernandez-Castro, Jun 29 2011

Some of the larger entries may only correspond to probable primes.

a(11) > 30000. - Michael S. Branicky, Apr 20 2025

			6! + 2*6 + 1 = 733 is prime, so 6 is in the sequence.

Cf. A073308 (k!+k+1 is prime).

[ n: n in [0..1000] | IsPrime(Factorial(n)+2*n+1) ]; // Klaus Brockhaus, Jul 02 2011

select(p -> isprime(factorial(p) + 2*p + 1), [$0 .. 1000]) # Reza K Ghazi, Jul 11 2021

Select[Range[0, 1000], PrimeQ[#! + 2*# + 1] &] (* Reza K Ghazi, Jul 11 2021 *)

for(n=0,10^5,N=n!+2*n+1;if(ispseudoprime(N),print1(n,", ")));
/* use isprime() for stricter checking */ /* Joerg Arndt, Jul 03 2011 */

from sympy import isprime
factor = 1;
for i in range(0, 10000):
    if i < 2:
        factor = 1
    else:
        factor = factor*i
    if isprime(factor + 2*i + 1):
        print(i, end=', ')

[p for p in range(1000) if is_prime(factorial(p)+2*p+1)] # Reza K Ghazi, Jul 11 2021

0 added by Arkadiusz Wesolowski, Jun 29 2011
a(9) from Arkadiusz Wesolowski, Jul 02 2011
a(10) from Charles R Greathouse IV, Oct 14 2011

A365639 Numbers k such that k! + k^3 + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 16, 28, 42

Offset: 1

Darío Clavijo, Sep 14 2023

If k is a term then k+1 is not composite (since k! + k^3 + 1 is divisible by k+1 for a composite k > 4). - Amiram Eldar, Sep 14 2023

Any further terms exceed 50000. - Lucas A. Brown, Mar 05 2024

Cf. A000040, A000142, A000578, A001093, A038507, A073308, A080668.

Select[Range[0, 50], ! CompositeQ[# + 1] && PrimeQ[#! + #^3 + 1] &] (* Amiram Eldar, Sep 14 2023 *)

from sympy import isprime, factorial
print([k for k in range(0, 43) if isprime((factorial(k)+ k**3 + 1))])

cp's OEIS Frontend

A090786 Least nonnegative integer k such that n! + n + k + 1 is prime.

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A073443 Numbers k such that k! - k - 1 is prime.

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

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A073309 Primes of the form k! + k + 1.

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A092791 Numbers k such that (k-1)! + k is prime.

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A192367 Numbers k such that k! + 2*k + 1 is prime.

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A365639 Numbers k such that k! + k^3 + 1 is prime.

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