A348153 - OEIS

%I #32 Oct 07 2021 13:57:16

%S 2,3,5,17,41,73,89,97,137,193,233,257,313,353,449,457,569,641,809,857,

%T 929,1033,1049,1097,1129,1153,1193,1217,1289,1409,1433,1601,1609,1697,

%U 1753,1889,1913,1993,2081,2137,2153,2273,2297,2393,2473,2617,2633,2657,2689,2713,2753,2777,2969

%N 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.

%C 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.

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

%t lim = 2000; p = 2; listc = {}; listp = {}; While[p < lim, n = 1;

%t While[n <= (p - 3)/4,

%t   If[PrimeQ[(p + 2 n)/(2 n + 1)], n = 2*p, n = n + 1]];

%t If[n == 2*p, AppendTo[listc, p]]; AppendTo[listp, p];

%t p = NextPrime[p]]; Complement[listp, listc]

%o (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

%Y Cf. A136020, A091180, A136061.

%K nonn

%O 1,1

%A _René Gy_, Oct 03 2021

%E More terms from _Michel Marcus_, Oct 04 2021