A358718 A sequence of sorted primes p_1 = 2, p_2 = 3, p_3 = 5, p_4 =7, p_5 < ... < p_m such that, for i >= 5, (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 p_1*p_2*...*p_(i-1).

2, 3, 5, 7, 11, 13, 19, 29, 37, 41, 43, 59, 73, 83, 109, 113, 131, 163, 173, 181, 227, 257, 331, 347, 353, 379, 419, 491, 523, 571, 601, 653, 661, 677, 739, 757, 769, 811, 859, 1091, 1201, 1217, 1297, 1307, 1321, 1459, 1481, 1621, 1721, 2029, 2081, 2089, 2179

Offset: 1

Author

Lorenzo Sillari, Nov 28 2022

Keywords

• nonn

Comments

The sequence was used, together with A358717 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.

Similar to A001259.

The sequence is a slight modification of A358717.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1000
• 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 A358717 and A358719.

Programs

Mathematica

s = {2, 3, 5, 7}; 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, 55] (* Amiram Eldar, Dec 01 2022 *)