A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

3, 4, 5, 7, 15, 22, 25, 26, 27, 35, 41, 47, 49, 50, 73, 74, 75, 87, 89, 95, 97, 98, 101, 107, 121, 122, 135, 145, 146, 147, 167, 193, 194, 195, 207, 215, 217, 218, 221, 227, 241, 242, 255, 275, 289, 290, 315, 327, 335, 337, 338, 347, 361, 362, 385, 386, 387, 395

Offset: 1

Author

Amiram Eldar, Dec 27 2019

Keywords

• nonn

Comments

Numbers of the form F(k) - 1 have the same Zeckendorf (A014417) and dual Zeckendorf (A104326) representations: alternating digits of 1 and 0 whose sum is floor((k - 1)/2). Thus, if k is in this sequence then F(k) - 1 is both a Zeckendorf-Niven number (A328208) and a lazy-Fibonacci-Niven number (A328212), i.e., A000071(a(n)) is in A330711.

Examples

			7 is in this sequence since F(7) - 1 = 13 - 1 = 12 is divisible by floor((7 - 1)/2) = 3. The Zeckendorf and dual Zeckendorf representations of 7 are both 1010, whose sum of digits, 2, divides 12. Thus 12 is both a Zeckendorf-Niven number and a lazy-Fibonacci-Niven number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A000071, A328208, A328212, A330711.

Programs

Mathematica

Select[Range[3, 400], Divisible[Fibonacci[#] - 1, Floor[(# - 1)/2]] &]