A090191 Asymmetric primes: an odd prime p is asymmetric if there is no odd prime q such that |p-q|=gcd(p-1,q-1).

23, 47, 83, 167, 173, 263, 359, 383, 389, 467, 479, 503, 509, 557, 563, 587, 653, 719, 797, 839, 863, 887, 907, 971, 983, 1103, 1187, 1259, 1283, 1307, 1367, 1439, 1499, 1511, 1523, 1571, 1579, 1637, 1733, 1823, 1907, 1913, 2039, 2063, 2099, 2203, 2207

Offset: 1

Author

Steven Finch, Jan 21 2004

Keywords

• nonn

Examples

			23 is asymmetric since gcd(22,q-1)=2<23-q for all odd primes q<23, gcd(22,22)=22>0 and gcd(22,q-1)=2<q-23 for all odd primes 23<q<67.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• P. Fletcher, W. Lindgren and C. Pomerance, Symmetric and asymmetric primes, J. Number Theory 58 (1996) 89-99.
• Carl Pomerance, Symmetric primes revisited, arXiv:1908.06161 [math.NT], 2019.

Crossrefs

Complement gives A090190.

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]]]; Complement[ Prime[ Range[2, 500]], Select[ Prime[ Range[2, 500]], f[ # ] != 0 &]] (* Robert G. Wilson v, Sep 19 2004 *)

PARI

is(n)=if(!isprime(n), return(0)); forprime(q=2,2*n, if(abs(n-q)==gcd(n-1,q-1), return(0))); 1 \\ Charles R Greathouse IV, Aug 08 2016

PARI

is(n)=if(!isprime(n), return(0)); fordiv(n\2, d, if(isprime(n-2*d) && gcd(n-1,n-2*d-1)==2*d, return(0)); if(isprime(n+2*d) && gcd(n-1,n+2*d-1)==2*d, return(0))); n>2 \\ Charles R Greathouse IV, Aug 08 2016

Extensions

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