A345899 - OEIS

%I #27 Nov 01 2024 15:42:49

%S 1,2,3,4,5,7,11,13,17,19,23,31,61,101,127,167,199,347

%N Exponents d of powers of 2, q = 2^d, such that each of q-1 and q+1 is either a power of prime or a semiprime.

%C a(19) > 1062, if it exists. - _Amiram Eldar_, Jun 29 2021

%H Peter Cameron, <a href="https://cameroncounts.wordpress.com/2020/10/07/between-fermat-and-mersenne/">Between Fermat and Mersenne</a>, Blog Post, October 07 2020.

%H Peter Cameron, <a href="/A345899/a345899.pdf">Between Fermat and Mersenne</a>, Blog Post, October 07 2020. [Local copy, with permission.]

%H Peter J. Cameron, Pallabi Manna, and Ranjit Mehatari, <a href="https://arxiv.org/abs/2106.14217">On finite groups whose power graph is a cograph</a>, arXiv:2106.14217 [math.GR], 2021. See Theorem 1.3 (b) pp. 3-4.

%e 2^13 = 8192, and 8191 is a prime and 8193 = 3*2731 is the product of twop primes, so 13 is a term. - _N. J. A. Sloane_, Nov 01 2024

%o (PARI) isor(q) = (q==1) || isprimepower(q) || (bigomega(q)==2);

%o isokb(d) = my(q=2^d); isor(q-1) && isor(q+1);

%Y Cf. A000961, A001358.

%Y Cf. also A345898, A345900.

%K nonn,hard,more

%O 1,2

%A _Michel Marcus_, Jun 29 2021

%E a(18) from _Amiram Eldar_, Jun 29 2021