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

1,1

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.

Links

Table of n, a(n) for n=1..53.

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]

Prog

(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

Crossrefs

Cf. A136020, A091180, A136061.

Sequence in context: A163588 A270539 A053182 * A211972 A339855 A076706

Adjacent sequences: A348150 A348151 A348152 * A348154 A348155 A348156

Keyword

nonn

Author

René Gy, Oct 03 2021

Extensions

More terms from Michel Marcus, Oct 04 2021

Status

approved