A090190 Symmetric primes: an odd prime p is symmetric if there exists an odd prime q such that |p-q| = gcd(p-1,q-1).

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Author

Steven Finch, Jan 21 2004

Keywords

• nonn

Examples

			Any twin prime is symmetric since 2=gcd(p-1,p+1) for any odd prime p.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• William Banks, Paul Pollack and Carl Pomerance, Symmetric primes revisited, arXiv:1908.06161 [math.NT], 2019.
• Rob Burns, Extremely symmetric primes, arXiv:2005.02922 [math.NT], 2020.
• P. Fletcher, W. Lindgren and C. Pomerance, Symmetric and asymmetric primes, J. Number Theory 58 (1996) 89-99.
• A. B. Kalmynin, On the Symmetry Graph of Prime Numbers, INTEGERS 21 (2021), #A2.

Crossrefs

Complement (in A000040) gives A090191.

Programs

Mathematica

f[n_] := Block[{k = 2}, While[k < 10^3 && Abs[n - Prime[k]] != GCD[n - 1, Prime[k] - 1], k++ ]; If[k == 10^3, 0, Prime[k]]]; Select[ Prime[ Range[2, 100]], f[ # ] != 0 &] (* Robert G. Wilson v, Sep 19 2004 *)

PARI

\\ Assumes n is an odd prime
is(p)=fordiv(p-1,d, if(isprime(p-d) || isprime(p+d), return(1))); 0 \\ Charles R Greathouse IV, Aug 24 2021

Extensions

More terms from Robert G. Wilson v, Sep 19 2004