cp's OEIS Frontend

A268923 All odd primes a(n) such that for all odd primes q smaller than a(n) the order of 2 modulo a(n)*q is a proper divisor of phi(a(n)*q)/2. The totient function phi is given in A000010.

%I A268923 #19 Sep 11 2024 22:59:27
%S A268923 17,31,41,43,73,89,97,109,113,127,137,151,157,193,223,229,233,241,251,
%T A268923 257,277,281,283,307,313,331,337,353,397,401,409,431,433,439,449,457,
%U A268923 499,521,569,571,577,593,601,617,631,641,643,673,683,691,727,733,739,761,769,809,811,857,881,911,919
%N A268923 All odd primes a(n) such that for all odd primes q smaller than a(n) the order of 2 modulo a(n)*q is a proper divisor of phi(a(n)*q)/2. The totient function phi is given in A000010.
%C A268923 This sequence was inspired by A269454 submitted by _Marina Ibrishimova_.
%C A268923 It seems that if for an odd prime p > 3 the order(2, p*3) < phi(p*3)/2 = p-1 then p is in this sequence.
%C A268923 Note that 2^(phi(p*q)/2) == 1 (mod p*q) for distinct odd primes p and q, due to Nagell's corollary on Theorem 64, p. 106. The products of distinct primes considered in the present sequence have order of 2 modulo p*q smaller than phi(p*q)/2.
%C A268923 Up to and including prime(100) = 541 the only odd primes p such that for all odd primes q smaller than p the order of 2 modulo p*q equals phi(p*q)/2 are 5, 7, and 11.
%C A268923 Complement of A216371 = A001122 U A105874 in the set of odd primes. Composed of the primes modulo which neither 2 nor -2 is a primitive root. Also, prime(n) is a term iff A376010(n) > 2. - _Max Alekseyev_, Sep 05 2024
%H A268923 Michael De Vlieger, <a href="/A268923/b268923.txt">Table of n, a(n) for n = 1..1000</a>
%H A268923 Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.
%e A268923 n=1: Order(2, 17*3) = 8, and 8 is a proper divisor of phi(17*3)/2 = 16;
%e A268923    order(2, 17*5) =  8, and 8 is a proper divisor of phi(17*5)/2 = 32;
%e A268923    order(2, 17*7) = 24, and 24 is a proper divisor of phi(17*7)/2 = 48;
%e A268923    order(2, 17*11) = 40, and 40 is a proper divisor of phi(17*11)/2 = 80;
%e A268923    order(2, 17*13) = 24, and 24 is a proper divisor of phi(17*13)/2 = 96.
%t A268923 Select[Prime@ Range[3, 157], Function[p, AllTrue[Prime@ Range[2, PrimePi@ p - 1], Function[q, With[{e = EulerPhi[p q]/2}, And[Divisible[e, #], # != e]] &@ MultiplicativeOrder[2, p q]]]]] (* _Michael De Vlieger_, Apr 01 2016, Version 10 *)
%Y A268923 Cf. A000010, A002326, A001122, A105874, A216371, A269454, A376010.
%K A268923 nonn
%O A268923 1,1
%A A268923 _Wolfdieter Lang_, Apr 01 2016
%E A268923 More terms from _Michael De Vlieger_, Apr 01 2016