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

%I A267999 #39 Dec 16 2022 09:35:52
%S A267999 35,55,77,95,115,119,143,155,161,187,203,209,215,221,235,247,253,275,
%T A267999 287,295,299,319,323,329,335,355,371,377,391,395,403,407,413,415,437,
%U A267999 455,473,475,493,497,515,517,527,533,535,539,551,559,575,581,583,589,611,623
%N A267999 Numbers n > 1 such that gcd(n, 2^n - 2) = 1.
%C A267999 Odd numbers n > 1 such that gcd(n, 2^(n-1)-1) = 1.
%C A267999 Conjecture: this is a subsequence of A121707. Tested for all terms <= 10^5.
%C A267999 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
%H A267999 Robert Israel, <a href="/A267999/b267999.txt">Table of n, a(n) for n = 1..10000</a>
%F A267999 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
%p A267999 select(n -> igcd(n, 2&^n-2 mod n)=1, [seq(i,i=3..10000, 2)]);
%t A267999 Select[Range[2, 768], GCD[#, 2^# - 2] == 1 &] (* or *)
%t A267999 Select[Range[2, 768], OddQ@ # && GCD[#, 2^(# - 1) - 1] == 1 &] (* _Michael De Vlieger_, Jan 24 2016 *)
%o A267999 (PARI) lista(nn) = for(n=2, nn, if(gcd(n, 2^n - 2) == 1, print1(n, ", "))); \\ _Altug Alkan_, Jan 24 2016
%o A267999 (Magma) [n: n in [2..800] | Gcd(n, 2^n-2) eq 1]; // _Vincenzo Librandi_, Jan 24 2016
%Y A267999 Cf. A121707.
%Y A267999 Cf. A306097 for terms of A121707 not in this sequence.
%K A267999 nonn
%O A267999 1,1
%A A267999 _Thomas Ordowski_ and _Robert Israel_, Jan 24 2016