A090190 -id:A090190 - cp's OEIS frontend

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

Steven Finch, Jan 21 2004

nonn

			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.

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.

Complement gives A090190.

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 *)

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

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

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

A275697 Primes p such that gcd(p-1, q-1) = q - p, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 41, 59, 61, 71, 73, 89, 97, 101, 107, 109, 113, 137, 149, 151, 157, 179, 181, 191, 193, 197, 227, 229, 239, 241, 269, 271, 277, 281, 311, 313, 331, 347, 349, 367, 373, 397, 401, 419, 421, 431, 433, 449, 457, 461, 521, 523

Offset: 1

Thomas Ordowski and Robert Israel, Aug 05 2016

nonn

Primes prime(k) such that prime(k) == 1 (mod A001223(k)).

Problem: are there infinitely many such primes?

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001223.
Contains A001359.
Except for 2, contained in A090190.

N:= 1000: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..nextprime(N),2)]):
P[select(i -> (P[i] - 1) mod (P[i+1]-P[i]) = 0, [$1..nops(P)-1])];

Select[Partition[Prime[Range[100]],2,1],GCD[#[[1]]-1,#[[2]]-1] == #[[2]]- #[[1]]&][[All,1]] (* Harvey P. Dale, Apr 18 2018 *)

is(n) = ispseudoprime(n) && gcd(n-1, nextprime(n+1)-1)==nextprime(n+1)-n \\ Felix Fröhlich, Aug 06 2016

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

A358312 Consider the graph of symmetric primes where p and q are connected if |p-q| = gcd(p-1,q-1). This sequence is an irregular table where the n-th row lists the first symmetric prime in a connected component with n vertices, with one representative for each nonisomorphic graph. Within a row, graphs are ordered by increasing size of its initial prime.

Original entry on oeis.org

3343, 42293, 461393, 70793, 72053, 268267, 8917219

Offset: 2

Charles R Greathouse IV, Nov 08 2022

Row lengths are A001349(n); if the sequence is finite the last row may be shorter.

Kalmynin gives T(2, 1) = 3343 and proves that, under a conjecture which is intermediate between Dickson's conjecture and the Bateman-Horn-Stemmler conjecture, that this sequence is infinite.

			T(2, 1) = 3343 has components {3343, 4457} which form the complete graph K_2.
T(3, 1) = 42293 has components {42293, 42487, 63439} which form the path graph P_3.
T(3, 2) = 461393 has components {461393, 519067, 692089} which form the complete graph K_3.
T(4, 1) = 70793 has components {70793, 106187, 106189, 123887} which form the claw graph.
T(4, 2) = 72053 has components {72053, 108079, 216157, 288209} which form the path graph P_4.
T(4, 3) = 268267 has components {268267, 357689, 536531, 536533} which form the paw graph.
T(4, 4) = 8917219 has components {8917219, 9908021, 14862031, 17834437} which form the square graph.

A. B. Kalmynin, On the Symmetry Graph of Prime Numbers, INTEGERS 21 (2021), #A2.

Cf. A090190.

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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).

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A275697 Primes p such that gcd(p-1, q-1) = q - p, where q is the next prime after p.

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

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