A267999 Numbers n > 1 such that gcd(n, 2^n - 2) = 1.

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581, 583, 589, 611, 623

Offset: 1

Author

Thomas Ordowski and Robert Israel, Jan 24 2016

Keywords

• nonn

Comments

Odd numbers n > 1 such that gcd(n, 2^(n-1)-1) = 1.

Conjecture: this is a subsequence of A121707. Tested for all terms <= 10^5.

For n > 1, if gcd(n, 2^n-2) = 1, then n is an "anti-Carmichael number" defined: p-1 does not divide n-1 for every prime p dividing n. Generally: for k > 1, gcd(k, b^k-b) = 1 for some integer b if and only if k is an "anti-Carmichael number". - Thomas Ordowski, Aug 14 2018

Links

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

Crossrefs

Cf. A121707.

Cf. A306097 for terms of A121707 not in this sequence.

Programs

Magma

[n: n in [2..800] | Gcd(n, 2^n-2) eq 1]; // Vincenzo Librandi, Jan 24 2016

Maple

select(n -> igcd(n, 2&^n-2 mod n)=1, [seq(i,i=3..10000, 2)]);

Mathematica

Select[Range[2, 768], GCD[#, 2^# - 2] == 1 &] (* or *)
Select[Range[2, 768], OddQ@ # && GCD[#, 2^(# - 1) - 1] == 1 &] (* Michael De Vlieger, Jan 24 2016 *)

PARI

lista(nn) = for(n=2, nn, if(gcd(n, 2^n - 2) == 1, print1(n, ", "))); \\ Altug Alkan, Jan 24 2016

Formula

a(n) = A121707(n) for n < 62. A121707(62) = 697 = A306097(1) is the first term of A121707 not in this sequence. - M. F. Hasler, Nov 09 2018