A204894 Least prime p such that n divides p-q for some prime q

3, 5, 5, 7, 7, 11, 17, 11, 11, 13, 13, 17, 29, 17, 17, 19, 19, 23, 41, 23, 23, 29, 53, 29, 53, 29, 29, 31, 31, 37, 67, 37, 71, 37, 37, 41, 79, 41, 41, 43, 43, 47, 89, 47, 47, 53, 97, 53, 101, 53, 53, 59, 109, 59, 113, 59, 59, 61, 61, 67, 127, 67, 131, 67, 67, 71

Offset: 1

Author

Clark Kimberling, Jan 20 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A204892.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Tony Haddad, Sun-Kai Leung, and Cihan Sabuncu, Visiting early at prime times, arXiv preprint (2024). arXiv:2408.11781 [math.NT]

Crossrefs

Cf. A204892, A204903, A000040.

Programs

Mathematica

(See the program at A204892.)

PARI

a(n)=forprime(p=n+2, , forstep(k=p%n, p-1, n, if(isprime(k), return(p)))) \\ Charles R Greathouse IV, Mar 20 2013

PARI

a(n)=if(isprime(n+2),return(n+2)); my(s=if(n%2,2*n,n),t); forprime(p=s+3,, t=p%n; forstep(q=if(t%2,t,t+n),p-s,s,if(isprime(q), return(p)))) \\ Charles R Greathouse IV, Jul 17 2015

PARI

a(n)=if(isprime(n+2),return(n+2)); my(s=if(n%2,2*n,n),r); forprime(p=s+3,2*s+1, if(isprime(p-s), return(p))); forprime(p=2*s+3,, r=p%n; forstep(q=if(r%2,r,r+n),p-s,s,if(isprime(q), return(p)))) \\ Charles R Greathouse IV, Aug 31 2024

Formula

n + 2 <= a(n) <= prime(n+1). - Charles R Greathouse IV, Jul 17 2015

Haddad, Leung, & Sabuncu prove that a(n) < 270*n for all large n. Probably this holds for all n. - Charles R Greathouse IV, Aug 29 2024