OFFSET
1,1
COMMENTS
All Mersenne primes >= 3 are terms (see A001348).
From Jianing Song, Oct 13 2023: (Start)
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.
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), ...
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)
LINKS
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Joerg Arndt, Sep 07 2015
EXTENSIONS
Corrected and extended by Giovanni Resta, Sep 09 2015
Terms a(17)-a(23) from Joerg Arndt, Sep 10 2015
STATUS
approved