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

3, 7, 31, 73, 85, 127, 2047, 3133, 4369, 8191, 11275, 49981, 60787, 76627, 121369, 131071, 140911, 178481, 262657, 486737, 524287, 599479, 1082401

Offset: 1

Author

Joerg Arndt, Sep 07 2015

Keywords

• nonn
• more

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

• Jianing Song, List of terms k of A261862 with ord(2,k) <= 47

Crossrefs

Cf. A261524, A001348.

Programs

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 *)

Extensions

Corrected and extended by Giovanni Resta, Sep 09 2015

Terms a(17)-a(23) from Joerg Arndt, Sep 10 2015