A090191 - OEIS

%I #22 Aug 20 2019 01:14:06

%S 23,47,83,167,173,263,359,383,389,467,479,503,509,557,563,587,653,719,

%T 797,839,863,887,907,971,983,1103,1187,1259,1283,1307,1367,1439,1499,

%U 1511,1523,1571,1579,1637,1733,1823,1907,1913,2039,2063,2099,2203,2207

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

%H Charles R Greathouse IV, <a href="/A090191/b090191.txt">Table of n, a(n) for n = 1..10000</a>

%H P. Fletcher, W. Lindgren and C. Pomerance, <a href="https://doi.org/10.1006/jnth.1996.0066">Symmetric and asymmetric primes</a>, J. Number Theory 58 (1996) 89-99.

%H Carl Pomerance, <a href="https://arxiv.org/abs/1908.06161">Symmetric primes revisited</a>, arXiv:1908.06161 [math.NT], 2019.

%e 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.

%t 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 *)

%o (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

%o (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

%Y Complement gives A090190.

%K nonn

%O 1,1

%A _Steven Finch_, Jan 21 2004

%E More terms from _Robert G. Wilson v_, Sep 19 2004