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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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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
EXAMPLE
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.
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 *)
PROG
(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
Primes p such that gcd(p-1, q-1) = q - p, where q is the next prime after p.
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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
COMMENTS
Primes prime(k) such that prime(k) == 1 (mod A001223(k)). Problem: are there infinitely many such primes?
MAPLE
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])];
MATHEMATICA
Select[Partition[Prime[Range[100]], 2, 1], GCD[#[[1]]-1, #[[2]]-1] == #[[2]]- #[[1]]&][[All, 1]] (* Harvey P. Dale, Apr 18 2018 *)
PROG
(PARI) is(n) = ispseudoprime(n) && gcd(n-1, nextprime(n+1)-1)==nextprime(n+1)-n \\ Felix Fröhlich, Aug 06 2016
CROSSREFS
Except for 2, contained in A090190.
Number of symmetric primes in the interval [prime(n)^2, prime(n)*prime(n+1)].
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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
COMMENTS
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.
EXAMPLE
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).
PROG
(PARI) issym(p) = fordiv(p-1, d, if(isprime(p-d) || isprime(p+d), return(1))); 0; \\ A090190a(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
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.
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3343, 42293, 461393, 70793, 72053, 268267, 8917219
COMMENTS
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.
EXAMPLE
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.
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