A134492 -id:A134492 - cp's OEIS frontend

A175130 Indices of Fibonacci numbers that are not cubefree.

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 125, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 250, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324

Offset: 1

R. J. Mathar, Feb 16 2010

nonn

Supersequence of A037917.
Conjecture: all terms are multiples of 6 or 125. - Harvey P. Dale, Apr 28 2020
The conjecture is false. The counterexamples are 392, 784, 1183, 1210, .... . - Amiram Eldar, Oct 16 2023

			Fibonacci(125) = 5^3 * 3001 * 158414167964045700001 = A000045(125) is not cubefree, which adds 125 to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..248
Blair Kelly, Fibonacci factorizations.

Cf. A000045, A010056, A037917, A046099, A134492, A212793.

import Data.List (findIndices)
a175130 n = a175130_list !! (n-1)
a175130_list = map (+ 1) $ findIndices ((== 0) . a212793) $ tail a000045_list
-- Reinhard Zumkeller, May 27 2012

Select[Range[350],Max[FactorInteger[Fibonacci[#]][[All,2]]]>2&] (* Harvey P. Dale, Apr 28 2020 *)

is(n)=n>5 && vecmax(factor(fibonacci(n))[,2])>2 \\ Charles R Greathouse IV, Nov 07 2014

A000045 INTERSECT A046099.

A010056(a(n)) * (1 - A212793(a(n))) = 1. - Reinhard Zumkeller, May 27 2012

A269500 a(n) = Fibonacci(10*n).

Original entry on oeis.org

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605

Offset: 0

Ilya Gutkovskiy, Mar 03 2016

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (123,-1)

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).

Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]

a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016

concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

G.f.: 55*x/(1 - 123*x + x^2).
a(n) = 123*a(n-1) - a(n-2).
a(n) = A000045(10*n).
Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…

A335976 Numbers k such that Fibonacci(6*k) is not a totient.

Original entry on oeis.org

0, 11, 13, 17, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 113, 121, 131, 137, 139, 141, 149, 151, 157, 167, 173, 191, 193, 199, 223, 229, 233, 239, 241, 243, 257, 263, 271, 281, 283, 293, 311, 313, 317, 321, 331, 339, 347, 349, 353, 373, 389, 397, 401, 419, 421, 431, 433, 443, 449, 457, 461, 479, 487, 509, 521, 541, 557, 573, 577, 587, 599, 613, 617, 619, 631, 641, 643, 653, 661, 673, 733, 739

Offset: 1

Altug Alkan, Jul 03 2020

nonn

Conjecture: Sequence contains infinitely many primes.

			11 is a term since Fibonacci(66) = 27777890035288 is not a totient number.

Cf. A000010, A007617, A134492, A280592, A280681.

PARI
```
isok(n) = !istotient(fibonacci(6*n))
```

a(12)-a(20) from Max Alekseyev, Aug 02 2020

Terms a(21) onward from Max Alekseyev, May 19 2024

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A175130 Indices of Fibonacci numbers that are not cubefree.

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A269500 a(n) = Fibonacci(10*n).

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A335976 Numbers k such that Fibonacci(6*k) is not a totient.

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