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A261862 Terms in A261524 that are not multiples of earlier terms.

%I A261862 #58 Jul 14 2024 20:24:11
%S A261862 3,7,31,73,85,127,2047,3133,4369,8191,11275,49981,60787,76627,121369,
%T A261862 131071,140911,178481,262657,486737,524287,599479,1082401
%N A261862 Terms in A261524 that are not multiples of earlier terms.
%C A261862 All Mersenne primes >= 3 are terms (see A001348).
%C A261862 From _Jianing Song_, Oct 13 2023: (Start)
%C A261862 In A261524 it is conjectured that degree(gcd( 1 + x^(Zs(d,2,1)), 1 + (1+x)^(Zs(d,2,1))) > 0 for every odd number d != 1, 15, 21, where Zs(d,2,1) is the d-th Zsigmondy number with parameters (2,1) (A064078). Since Zsigmondy numbers with different indices are coprime, if this conjecture is true, then there exists a term of this sequence k with ord(2,k) = d, and k must be a divisor of Zs(d,2,1) for every odd number d != 1, 15, 21. Here ord(a,k) is the multiplicative order of 2 modulo k. In A261524 we show that this conjecture is true for powers > 1 of a prime r >= 5, so there are infinitely many terms in this sequence.
%C A261862 One may conjecture that, if k is a term with ord(2,k) = d for even d, then k is a divisor of Zs(d,2,1)*Zs(d/2,2,1). This fails for (d,k) = (20,11275), (40,16962275), (44,165965585), ...
%C A261862 Conjecture: a term with ord(2,k) = d for even d exists if and only if d != 12 or 2*p, where p is any Mersonne exponent. (End)
%H A261862 Jianing Song, <a href="/A261862/a261862.txt">List of terms k of A261862 with ord(2,k) <= 47</a>
%t A261862 n=1; t= L= {}; While[n<5000, n+=2; If[ CoefficientList[ PolynomialGCD[1 + x^n, 1 + (x + 1)^n, Modulus->2], x] !={1}, If[ Intersection[Divisors@ n, t] == {}, Print@ AppendTo[L, n]]; AppendTo[t, n]]]; L (* _Giovanni Resta_, Sep 07 2015 *)
%Y A261862 Cf. A261524, A001348.
%K A261862 nonn,more
%O A261862 1,1
%A A261862 _Joerg Arndt_, Sep 07 2015
%E A261862 Corrected and extended by _Giovanni Resta_, Sep 09 2015
%E A261862 Terms a(17)-a(23) from _Joerg Arndt_, Sep 10 2015