A358718 - OEIS

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).

%I #19 Feb 20 2024 15:30:49

%S 2,3,5,7,11,13,19,29,37,41,43,59,73,83,109,113,131,163,173,181,227,

%T 257,331,347,353,379,419,491,523,571,601,653,661,677,739,757,769,811,

%U 859,1091,1201,1217,1297,1307,1321,1459,1481,1621,1721,2029,2081,2089,2179

%N 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).

%C 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.

%C Similar to A001259.

%C The sequence is a slight modification of A358717.

%H Max Alekseyev, <a href="/A358718/b358718.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Ferrari and L. Sillari, <a href="https://arxiv.org/abs/2110.05401">On the minimal number of solutions of the equation phi(n+k) = M*phi(n), M=1,2</a>, arXiv:2110.05401 [math.NT], 2021.

%t 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 *)

%Y Similar to A001259. See also A358717 and A358719.

%K nonn

%O 1,1

%A _Lorenzo Sillari_, Nov 28 2022