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Indices of Fibonacci numbers that are not cubefree.
1

%I #22 Oct 16 2023 04:33:02

%S 6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96,102,108,114,120,125,

%T 126,132,138,144,150,156,162,168,174,180,186,192,198,204,210,216,222,

%U 228,234,240,246,250,252,258,264,270,276,282,288,294,300,306,312,318,324

%N Indices of Fibonacci numbers that are not cubefree.

%C Supersequence of A037917.

%C Conjecture: all terms are multiples of 6 or 125. - _Harvey P. Dale_, Apr 28 2020

%C The conjecture is false. The counterexamples are 392, 784, 1183, 1210, .... . - _Amiram Eldar_, Oct 16 2023

%H Amiram Eldar, <a href="/A175130/b175130.txt">Table of n, a(n) for n = 1..248</a>

%H Blair Kelly, <a href="http://mersennus.net/fibonacci//f1000.txt">Fibonacci factorizations</a>.

%F A000045 INTERSECT A046099.

%F A010056(a(n)) * (1 - A212793(a(n))) = 1. - _Reinhard Zumkeller_, May 27 2012

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

%t Select[Range[350],Max[FactorInteger[Fibonacci[#]][[All,2]]]>2&] (* _Harvey P. Dale_, Apr 28 2020 *)

%o (Haskell)

%o import Data.List (findIndices)

%o a175130 n = a175130_list !! (n-1)

%o a175130_list = map (+ 1) $ findIndices ((== 0) . a212793) $ tail a000045_list

%o -- _Reinhard Zumkeller_, May 27 2012

%o (PARI) is(n)=n>5 && vecmax(factor(fibonacci(n))[,2])>2 \\ _Charles R Greathouse IV_, Nov 07 2014

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

%K nonn

%O 1,1

%A _R. J. Mathar_, Feb 16 2010