A106628 Anomalous prime numbers.

199, 211, 283, 317, 337, 389, 491, 509, 547, 577, 619, 683, 701, 773, 787, 797, 863, 887, 1069, 1109, 1129, 1153, 1163, 1373, 1381, 1409, 1459, 1523, 1531, 1571, 1627, 1637, 1669, 1709, 1723, 1733, 1759, 1831, 1889, 1913, 1933, 1951, 1979, 2003, 2017

Offset: 1

Author

Pahikkala Jussi, May 11 2005

Keywords

• base
• nonn
• changed

Comments

If x and y are two consecutive prime numbers (x < y), Euclid's algorithm gives integers t and d such that tx+dy = 1 = gcd(x, y). The algorithm "Anomalia" gives t and d such that |t+d| is as small as possible (it is often = 1). The prime number x is 'anomalous' iff |t+d| > 1 for x and y.

That is, primes p such that neither q-1 nor q+1 is divisible by q-p, where q is the next prime larger than p. - Charles R Greathouse IV, Aug 20 2017

Examples

			a(1) = 199 because -88*199+83*211 = 1, |-88+83| = 5 > 1;
|tx+dy| = 1 for all primes x < 199 (when t and d are determined by the algorithm "Anomalia")
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A083371.

Programs

Mathematica

q[x_]:=Module[{y,d},If[!PrimeQ[x],Return[0]];y=NextPrime[x + 1];d=y-x;Mod[y-1, d]!=0 && Mod[y+1, d] != 0];Select[Range[2017],q] (* James C. McMahon, Aug 23 2025 *)

PARI

is(x)=if(!isprime(x), return(0)); my(y=nextprime(x+1),d=y-x); (y-1)%d && (y+1)%d \\ Charles R Greathouse IV, Aug 20 2017

Formula

Conjecture: a(n) ~ n log n. - Charles R Greathouse IV, Aug 20 2017