A358717 A sequence of sorted primes 2 = p_1 < p_2 < ... < p_m such that (p_i + 1)/2 divides the product p_1*p_2*...*p_(i-1) of the earlier primes and each prime factor of (p_i-1)/2 is a prime factor of the product.

2, 3, 5, 11, 19, 37, 73, 109, 1459, 2179, 2917, 4357, 8713

Offset: 1

Author

Lorenzo Sillari, Nov 28 2022

Keywords

• nonn
• full
• fini

Comments

The sequence was used, together with A358718 and A358719, by Ferrari and Sillari (Preprint-2022) to prove that there are at least three solutions n to phi(n+k) = 2* phi(n) for all even k <= 4*10^58.

I have checked up to 10^8 and found no more terms.

Prime a(14) does not exist, which can be established by going over the divisors d of the product a(1)*...*a(13) and testing 2*d-1 as a candidate for a(14). - Max Alekseyev, Feb 19 2024

Links

• M. Ferrari and L. Sillari, On the minimal number of solutions of the equation phi(n+k) = M*phi(n), M=1,2, arXiv:2110.05401 [math.NT], 2021.

Crossrefs

Similar to A001259.

See also A358718 and A358719.

Programs

Mathematica

s = {2}; step[s_] := Module[{p = NextPrime[s[[-1]]], r = Times @@ s}, While[! Divisible[r, (p + 1)/2] || ! Divisible[r, Times @@ FactorInteger[(p - 1)/2][[;; , 1]]], p = NextPrime[p]]; Join[s, {p}]]; Nest[step, s, 12] (* Amiram Eldar, Nov 30 2022 *)

Extensions

Keywords 'full' and 'fini' added by Max Alekseyev, Feb 19 2024