A348153 Primes for which there is no pair (k,q) with k a positive integer and q another prime, such that p=q*(2k+1)-2k.

2, 3, 5, 17, 41, 73, 89, 97, 137, 193, 233, 257, 313, 353, 449, 457, 569, 641, 809, 857, 929, 1033, 1049, 1097, 1129, 1153, 1193, 1217, 1289, 1409, 1433, 1601, 1609, 1697, 1753, 1889, 1913, 1993, 2081, 2137, 2153, 2273, 2297, 2393, 2473, 2617, 2633, 2657, 2689, 2713, 2753, 2777, 2969

Offset: 1

Author

René Gy, Oct 03 2021

Keywords

• nonn

Comments

There are primes p for which there exist a positive integer k and another prime q such that p=q*(2k+1)-2k. See A136020, A091180, A136061 and the subsequent sequences. Such k is called an "order" of the prime p. Note that q is necessarily larger than 2 and that 4*k is necessarily smaller than p-1. A prime may belong to more than one order, but the primes listed in the present sequence do not belong to any order.

As soon as they are larger than 8, all members minus 1 are multiples of 8.

Crossrefs

Cf. A136020, A091180, A136061.

Programs

Mathematica

lim = 2000; p = 2; listc = {}; listp = {}; While[p < lim, n = 1;
While[n <= (p - 3)/4,
  If[PrimeQ[(p + 2 n)/(2 n + 1)], n = 2*p, n = n + 1]];
If[n == 2*p, AppendTo[listc, p]]; AppendTo[listp, p];
p = NextPrime[p]]; Complement[listp, listc]

PARI

isok(p) = {if (isprime(p), for (k=1, (p-3)/4, my(q = (p+2*k)/(2*k+1)); if ((denominator(q)==1) && isprime(q), return(0));); return (1););} \\ Michel Marcus, Oct 07 2021

Extensions

More terms from Michel Marcus, Oct 04 2021