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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). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A248077 A239547 A060728 * A295988 A216433 A101761

Adjacent sequences:  A330708 A330710 A330711 * A330713 A330714 A330715

KEYWORD

nonn

AUTHOR

Amiram Eldar, Dec 27 2019

STATUS

approved

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Last modified July 4 16:24 EDT 2020. Contains 335448 sequences. (Running on oeis4.)