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.

17, 31, 41, 43, 73, 89, 97, 109, 113, 127, 137, 151, 157, 193, 223, 229, 233, 241, 251, 257, 277, 281, 283, 307, 313, 331, 337, 353, 397, 401, 409, 431, 433, 439, 449, 457, 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

Offset: 1

Author

Wolfdieter Lang, Apr 01 2016

Keywords

• nonn

Comments

This sequence was inspired by A269454 submitted by Marina Ibrishimova.

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.

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.

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.

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

Examples

			n=1: Order(2, 17*3) = 8, and 8 is a proper divisor of phi(17*3)/2 = 16;
   order(2, 17*5) =  8, and 8 is a proper divisor of phi(17*5)/2 = 32;
   order(2, 17*7) = 24, and 24 is a proper divisor of phi(17*7)/2 = 48;
   order(2, 17*11) = 40, and 40 is a proper divisor of phi(17*11)/2 = 80;
   order(2, 17*13) = 24, and 24 is a proper divisor of phi(17*13)/2 = 96.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1000
• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Crossrefs

Cf. A000010, A002326, A001122, A105874, A216371, A269454, A376010.

Programs

Mathematica

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 *)

Extensions

More terms from Michael De Vlieger, Apr 01 2016