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

1,1

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.

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.

Example

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.

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 *)

Crossrefs

Cf. A000010, A002326, A269454.

Sequence in context: A095748 A350699 A235920 * A172287 A062579 A134076

Adjacent sequences: A268920 A268921 A268922 * A268924 A268925 A268926

Keyword

nonn

Author

Wolfdieter Lang, Apr 01 2016

Extensions

More terms from Michael De Vlieger, Apr 01 2016

Status

approved