A087202 -id:A087202 - cp's OEIS frontend

A037153 a(n) = p-n!, where p is the smallest prime > n!+1.

2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 47, 67, 223, 107, 127, 79, 37, 97, 61, 131, 311, 43, 97, 53, 61, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271, 67, 193

Offset: 1

Jud McCranie

nonn

Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 1200 terms are primes. Are all terms prime?
a(n) is the first (smallest) m such that m > 1 and n!+ m is prime. The second such m is A087202(n). a(n) must be greater than nextprime(n)-1. - Farideh Firoozbakht, Sep 01 2003
Sequence A069941, which counts the primes between n! and n!+n^2, provides numerical evidence that the smallest prime p greater than n!+1 is a prime distance from n!; that is, p-n! is a prime number. For p-n! to be a composite number, p would have to be greater than n!+n^2, which would imply that A069941(n)=0. - T. D. Noe, Mar 06 2010
The first 4003 terms are prime. - Dana Jacobsen, May 10 2015

Ray Chandler and Dana Jacobsen, Table of n, a(n) for n = 1..4000 [first 1200 terms from Ray Chandler]
Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
Hisanori Mishima, Primes near to factorial, Dec 2008.
Andy Nicol, Line graphs of A037153 in order and ascending numerical order

Cf. A087202, A005235, A033932.

z:=125; [p-f where p is NextPrime(f+1) where f is Factorial(n): n in [1..z]]; // Klaus Brockhaus, Mar 02 2010

NextPrime[ n_Integer ] := (k=n+1; While[ !PrimeQ[ k ], k++ ]; Return[ k ]); f[ n_Integer ] := (p = n! + 1; q = NextPrime[ p ]; Return[ q - p + 1 ]); Table[ f[ n ], {n, 1, 75} ] (* Robert G. Wilson v *)

for n from 1 to 65 do f := n!:a := nextprime(f+2)-f:print(a) end_for; // Zerinvary Lajos, Feb 22 2007

a(n)=nextprime(n!+2)-n! \\ Charles R Greathouse IV, Jul 02 2013; Corrected by Dana Jacobsen, May 10 2015

use ntheory ":all"; for my $n (1..1000) { my $f=factorial($n); say "$n ",next_prime($f+1)-$f; } # Dana Jacobsen, May 10 2015

from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(fn+1) - fn
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, May 22 2022

Edited by N. J. A. Sloane, Mar 06 2010

A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

Original entry on oeis.org

2, 2, 3, 7, 8, 15, 8, 18, 16, 19, 12, 20, 11, 8, 11, 6, 12, 23, 24, 15, 31, 21, 27, 15, 16, 26, 25, 17, 17, 29, 20, 27, 27, 30, 23, 16, 28, 23, 25, 29, 15, 24, 19, 36, 36, 39, 15, 36, 24, 44, 35, 29, 27, 25, 36, 22, 37, 31, 32, 41, 29, 55, 27, 45, 29, 59, 34, 37, 24, 49, 25, 40

Offset: 0

Frederick Magata (frederick.magata(AT)web.de), May 13 2009

nonn

The conjectures from A037153 and A087202 can be rephrased using a(n):
Is a(n)>=2 for all n>=0 and a(n)>=3 for all n>=2?
Also compare this with the conjecture on the fortunate numbers A005235.
Is the following true: for every m there is an N such that for all n>N a(n)>m?
There even seems to be the estimate a(n)>log(n+1)*sqrt(n+1)/2.

			a(3)=7: The seven primes following 3!+1=7 are 11,13,17,19,23,29 and 31.
Subtracting 3!=6 from each of them gives 5,7,11,13,17,23 and 25.
The first six values are prime, while the seventh 25=5^2 is not.

Cf. A005235, A037153, A087202.

a:=proc(n) option remember; local k:
for k from 1 while isprime((nextprime@@k)(n!+1)-n!) do od:
k; end;

A087203 a(n) is the smallest m such that m > A037155(n) and n!- m is prime.

Original entry on oeis.org

4, 7, 11, 19, 19, 37, 17, 17, 17, 17, 61, 43, 59, 71, 61, 43, 113, 71, 41, 101, 191, 103, 191, 179, 71, 127, 37, 97, 113, 373, 71, 373, 293, 157, 149, 241, 167, 211, 151, 89, 131, 113, 73, 107, 179, 227, 173, 113, 257, 239, 151, 227, 163, 509, 293, 347, 643, 373, 457

Offset: 3

Farideh Firoozbakht, Sep 01 2003

nonn

a(1) and a(2) are not defined. a(n) is the second m (first m is A037155(n)) such that m > 1 and n!- m is prime.For 3 < n < 643,a(n) is prime. I guess (compare the conjecture about A087202) except for the first term, every term of this sequence is prime.

Cf. A037155, A087201, A087202.

A037155[3]=3; A037155[n_] := (For[m=Prime[PrimePi[n]+1], !PrimeQ[n!-m], m++ ]; m); a[n_] := (For[m=A037155[n]+1, !PrimeQ[n!-m], m++ ]; m); Table[a[n], {n, 3, 62}]

A037155[3]=3; A037155[n_] := (For[m=Prime[PrimePi[n]+1], !PrimeQ[n!-m], m++ ]; m); a[n_] := (For[m=A037155[n]+1, !PrimeQ[n!-m], m++ ]; m)

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A037153 a(n) = p-n!, where p is the smallest prime > n!+1.

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A160433 a(n) is the least number k such that (k-th prime after n!+1)-n! is not a prime.

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A087203 a(n) is the smallest m such that m > A037155(n) and n!- m is prime.

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