A187063 Numbers of the form (4^k - 1)/3 whose greatest prime divisor is of the form 2^q - 1 or 2^q + 1.

5, 21, 85, 341, 5461, 21845, 22369621, 89478485, 1431655765, 5726623061, 91625968981, 1501199875790165, 1537228672809129301, 98382635059784275285, 1690200800304305868662270940501, 1772303994379887830538409413707126101

Offset: 1

Author

Michel Lagneau, Mar 03 2011

Keywords

• nonn

Comments

The binary expansion of (4^k-1)/3 has no consecutive equal binary digits.

The corresponding values of k are 2, 3, 4, 5, 7, 8, 13, 14, 16, 17, 19, 26, 31, 34, 51, 61, 62, 89, 107, 122, 127, 178, 214, 254, 521, ... - Amiram Eldar, Mar 02 2020

Examples

			(4^6-1)/3 = 1365 = 3 * 5 * 7 * 13 is not in the sequence because  13 is not of the form 2^q +/- 1 ;
(4^16-1)/3 = 1431655765 = 5 * 17 * 257 * 65537 and 65537 = 2^16 + 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..25

Crossrefs

Cf. A002450 ((4^n-1)/3), A274906.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local k, t, d, h;
      for k from 1+ `if`(n=1, 0, ilog[4](a(n-1)*3+1))
      do t:= (4^k-1)/3;
         d:= max(factorset(t)[]);
         for h in [d+1, d-1] do
            if 2^ilog[2](h)=h then RETURN(t) fi
         od
      od
    end:
seq(a(n), n=1..17);  # Alois P. Heinz, Mar 04 2011

Mathematica

okQ[n_] := Module[{p = FactorInteger[n][[-1, 1]]}, IntegerQ[Log[2, p + 1]] || IntegerQ[Log[2, p - 1]]]; t = Table[(4^n-1)/3, {n,2,50}]; Select[t, okQ] (* T. D. Noe, Mar 04 2011 *)