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Search: id:A065069
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| A065069 |
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Numbers n such that Fibonacci(n) is not square-free, but for all proper divisors k of n, Fibonacci(k) is square-free. |
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3
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| 6, 25, 56, 91, 110, 153, 406, 703, 752, 820, 915, 979, 1431, 1892, 2147, 2701, 2943, 3029, 3422, 4378, 4556, 4753, 4970, 5513, 6394, 7868, 8841, 9453, 10712, 12403, 13508, 13546, 15051, 16256, 17030, 17267, 18023, 18721, 19503, 20827, 21206
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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These are first primitive indices m for which Fib(m) is squareful. Note that Fib(km) is divisible by Fib(m).
This sequence is closely related to A001602(n), which gives the index of the smallest Fibonacci number divisible by prime(n). It can be shown that the first Fibonacci number divisible by prime(n)^2 is A001602(n)*prime(n). This sequence is the collection of numbers A001602(n)*prime(n) with multiples removed. For example, A001602(2)*prime(2) = 12, but all multiples of 12 will generated by 6, the first number in this sequence. The Mathematica code assumes that Fibonacci numbers do not have any square primitive prime factors -- an assumption whose truth is an open question. - T. D. Noe (noe(AT)sspectra.com), Jul 24 2003
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LINKS
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Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Blair Kelly, Fibonacci and Lucas Factorizations
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EXAMPLE
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a(1) = 6 because 2^2 divides Fibonacci(6) but no smaller Fibonacci number.
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MATHEMATICA
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<< NumberTheory`NumberTheoryFunctions`; a = {}; l = 0; Do[m = n; If[k = 1; While[k < l + 1 && !IntegerQ[ n/ a[[k]]], k++ ]; k > l, If[ !SquareFreeQ[ Fibonacci[n]], a = Append[a, n]; l++; Print[n]]], {n, 1, 480} ]
nLimit=50000; i=3; pMax=1; iMax=1; While[p=Transpose[FactorInteger[Fibonacci[i]]][[1, -1]]; i*p<nLimit, If[p>pMax, pMax=p; iMax=i]; i++ ]; nMax=PrimePi[pMax]; fs={}; Do[p=Prime[n]; k=1; found=False; While[found=(Mod[Fibonacci[k], p]==0); !found&&k*p<nLimit, k++ ]; If[found, AppendTo[fs, p*k]], {n, nMax}]; fs=Sort[fs]; i=1; While[i<=Length[fs]-1, n=fs[[i]]; If[n>0, j=i+1; While[j<=Length[fs], If[Mod[fs[[j]], n]==0, fs[[j]]=0]; j++ ]]; i++ ]; Select[fs, #>0&&#<nLimit&]
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CROSSREFS
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A037917 (all indices <=240 for which Fib(m) is squareful)
Cf. A065106, A001602, A013929 (not square-free).
Sequence in context: A067926 A075224 A042185 this_sequence A022270 A001664 A096958
Adjacent sequences: A065066 A065067 A065068 this_sequence A065070 A065071 A065072
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KEYWORD
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nonn
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AUTHOR
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Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 07 2001
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EXTENSIONS
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One more terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 08 2001
More terms from T. D. Noe (noe(AT)sspectra.com), Jul 24 2003
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