A269454 -id:A269454 - 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.

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

Wolfdieter Lang, Apr 01 2016

nonn

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

			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.

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.

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

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 *)

More terms from Michael De Vlieger, Apr 01 2016

A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

Original entry on oeis.org

24, 40, 60, 88, 132, 184, 220, 232, 276, 348, 328, 460, 424, 492, 580, 636, 664, 712, 820, 1012, 904, 996, 1060, 1068, 1048, 1276, 1356, 1384, 1432, 1660, 1572, 1804, 1528, 1780, 1864, 1912, 2076, 2332, 2260, 2148, 2008, 2292, 2668, 2248, 2620, 2344, 2796, 2868, 3012, 2872, 3460, 3652, 3772, 3372

Offset: 1

Marina Ibrishimova, Feb 27 2016

nonn

phi(p*q) = (p-1)(q-1) where p, q are distinct safe primes.

2^(a(n)/2) == 1 (mod A157352(n)). For the reference see a comment on A269454. - Wolfdieter Lang, Mar 31 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Cf. A000010, A157352, A269454.

EulerPhi /@ Select[Select[Range@ 4000, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)

a(n) = phi(A157352(n)), n >= 1.

More terms from Michael De Vlieger, Feb 28 2016

A294091 Numbers k such that (k - 1)/2 is prime that are not congruent to -1 mod 8.

Original entry on oeis.org

5, 11, 27, 35, 59, 75, 83, 107, 123, 147, 179, 195, 203, 219, 227, 275, 299, 315, 347, 363, 387, 395, 459, 467, 483, 515, 539, 555, 563, 587, 627, 635, 675, 699, 707, 747, 779, 795, 803, 819, 843, 867, 899, 915, 923, 1019, 1043, 1083, 1115, 1139, 1155, 1187

Offset: 1

Arkadiusz Wesolowski, Feb 09 2018

k is prime if and only if 2^(k - 1) is congruent to 1 mod k. The test relies on the Selfridge criterion (see p. 42 of the Krizek et al. reference).

M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001, p. 42.
P. Ribenboim, The Little Book of Bigger Primes, Springer Science & Business Media, 2013, pp. 32-33.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005385. A269454 gives the primes.

map(t->2*t+1, select(isprime, [2,seq(4*k+1,k=1..1000)])); # Robert Israel, Feb 27 2018

Select[2*Prime@Range[109] + 1, ! Mod[#, 8] == 7 &]

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A294091 Numbers k such that (k - 1)/2 is prime that are not congruent to -1 mod 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A294091 Numbers k such that (k - 1)/2 is prime that are not congruent to -1 mod 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

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.