A090191 -id:A090191 - cp's OEIS frontend

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

Steven Finch, Jan 21 2004

nonn

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

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.

Complement (in A000040) gives A090191.

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 *)

\\ 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

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

A174222 Number of symmetric primes in the interval [prime(n)^2, prime(n)*prime(n+1)].

Original entry on oeis.org

1, 2, 2, 6, 4, 7, 5, 10, 18, 6, 24, 18, 10, 21, 35, 29, 14, 33, 27, 14, 44, 32, 43, 64, 36, 16, 36, 17, 38, 133, 41, 71, 16, 123, 21, 71, 72, 49, 90, 85, 36, 158, 34, 66, 31, 190, 184, 73, 39, 73, 109, 33, 188, 109, 117, 110, 35, 126, 85, 36, 221, 298, 99, 41, 95, 320, 136, 237

Offset: 1

Jaspal Singh Cheema, Mar 18 2010

nonn

If you graph a(n) vs. n, an interesting pattern with random-looking fluctuations begins to emerge.
As you go farther along the n-axis, greater are the number of symmetric primes, on average.
The smallest count of a(.)=1 occurs only once at the very beginning.
I suspect all a(n) are > 0. If one could prove this, it would imply that Symmetric primes are infinite.

			The square of the first prime is 2^2=4 and the product of the first and second prime is 2*3=6. Within this interval, there is 1 symmetric prime, which is 5. Hence a(1)=1.
The second term, a(2)=2, refers to the two symmetric primes 11 and 13 within the interval (9,15).

Cf. A090190, A090191.

Cf. A006094, A001248,

issym(p) = fordiv(p-1, d, if(isprime(p-d) || isprime(p+d), return(1))); 0; \\ A090190
a(n) = my(p=prime(n), nb=0); forprime(q=p^2, p*nextprime(p+1), if (issym(q), nb++)); nb; \\ Michel Marcus, Nov 03 2022

#{ A090190(j): A001248(n) < A090190(j) < A006094(n)}.

Edited by R. J. Mathar, Mar 31 2010

cp's OEIS Frontend

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).

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