A054495 -id:A054495 - cp's OEIS frontend

A054494 Largest Fibonacci factor of n.

1, 2, 3, 2, 5, 3, 1, 8, 3, 5, 1, 3, 13, 2, 5, 8, 1, 3, 1, 5, 21, 2, 1, 8, 5, 13, 3, 2, 1, 5, 1, 8, 3, 34, 5, 3, 1, 2, 13, 8, 1, 21, 1, 2, 5, 2, 1, 8, 1, 5, 3, 13, 1, 3, 55, 8, 3, 2, 1, 5, 1, 2, 21, 8, 13, 3, 1, 34, 3, 5, 1, 8, 1, 2, 5, 2, 1, 13, 1, 8, 3, 2, 1, 21, 5, 2, 3, 8, 89, 5, 13, 2, 3, 2, 5, 8, 1, 2, 3, 5

Offset: 1

Henry Bottomley, Apr 04 2000

			a(10)=5 because 1, 2 and 5 are the Fibonacci numbers which divide 10 and 5 is the largest.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A005086, A010056, A054495.
Sequences with similar definitions: A047930 (smallest Fibonacci multiple), A280686 (restricted to proper divisors), A280694 (equivalent for Lucas numbers).
Positions of 1's: A147956.

With[{fibs=Fibonacci[Range[20]]},Table[Max[Select[fibs,Divisible[ n,#]&]],{n,100}]] (* Harvey P. Dale, Jul 17 2012 *)

A010056(n)=my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))
a(n)=fordiv(n,d,if(A010056(n/d), return(n/d))) \\ Charles R Greathouse IV, Nov 05 2014

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A054494(n): return next(d for d in sorted(divisors(n,generator=True),reverse=True) if is_square(m:=5*d**2-4) or is_square(m+8)) # Chai Wah Wu, May 06 2024

a(n) = n/A054495(n).

Corrected by Harvey P. Dale, Jul 17 2012

A249783 Smallest index of a Fibonacci-like sequence containing n, see comments.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 3, 4, 1, 4, 3, 2, 5, 3, 5, 4, 1, 6, 4, 3, 5, 2, 7, 5, 3, 6, 5, 4, 6, 1, 7, 6, 4, 7, 3, 5, 7, 2, 8, 7, 5, 8, 3, 6, 8, 5, 9, 4, 6, 9, 1, 7, 9, 6, 10, 4, 7, 10, 3, 8, 5, 7, 11, 2, 8, 11, 7, 9, 5, 8, 12, 3, 9, 6, 8, 10, 5, 9, 13, 4, 10, 6, 9, 11, 1, 10, 7, 9, 11, 6, 10, 12, 4, 11, 7, 10

Offset: 0

Allan C. Wechsler, Nov 05 2014

Any two nonnegative integers F0 and F1 generate a Fibonacci-like sequence where for n > 1, Fn = F[n-1] + F[n-2]. Call F0 + F1 the "index" of such a sequence. In this sequence, a(n) is the smallest occurring index of any Fibonacci-like sequence containing n.

			For n = 0, the trivial sequence 0, 0, 0, ... has index 0.
For n = 5, the classic Fibonacci sequence 0, 1, 1, 2, 3, 5, ... contains 5 and has index 1.
For n = 7, the Lucas sequence 2, 1, 3, 4, 7, ... contains 7, and no such sequence with a smaller index contains 7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index to sequences related to the complexity of n

If n > 0 is an element of A000045, a(n) = 1. If n > 2 is twice such an element, a(n) = 2. If n > 3 is an element of A000032 or of A022086, a(n) = 3.

Haskell
```
bi x y = if (x
				
```

a[n_] := Module[{a, k, A, B}, If[n<2, Return[n]]; For[k=1, k <= n-1, k++, For[a=0, a <= k-1, a++, A=a; B=k-A; While[BJean-François Alcover, Jan 06 2017, translated from PARI *)

a(n)=if(n<2,return(n));for(k=1,n-1,for(a=0,k-1,my(A=a,B=k-A);while(BCharles R Greathouse IV, Nov 06 2014

a(n) <= A054495(n) <= n. - Charles R Greathouse IV, Nov 06 2014

Extended by Charles R Greathouse IV, Nov 06 2014

A280695 a(n) = n / A280694(n); n divided by the largest Lucas number (A000032) dividing n.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 2, 3, 5, 1, 3, 13, 2, 5, 4, 17, 1, 19, 5, 3, 2, 23, 6, 25, 13, 9, 4, 1, 10, 31, 8, 3, 17, 5, 2, 37, 19, 13, 10, 41, 6, 43, 4, 15, 23, 1, 12, 7, 25, 17, 13, 53, 3, 5, 8, 19, 2, 59, 15, 61, 31, 9, 16, 65, 6, 67, 17, 23, 10, 71, 4, 73, 37, 25, 1, 7, 26, 79, 20, 27, 41, 83, 12, 85, 43, 3, 8, 89, 5, 13, 23, 31, 2, 95, 24, 97, 14, 9, 25, 101, 34

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15127

Cf. A054495, A280694, A280697.

Scheme
```
(define (A280695 n) (/ n (A280694 n)))
```

A246399 Least k such that A249783(k) = n.

Original entry on oeis.org

1, 4, 7, 12, 17, 22, 27, 43, 51, 59, 67, 75, 83, 122, 135, 148, 161, 174, 187, 200, 213, 226, 239, 344, 365, 386, 407, 428, 449, 470, 491, 512, 533, 554, 575, 596, 617, 638, 659, 931, 965, 999, 1033, 1067, 1101, 1135, 1169, 1203, 1237, 1271

Offset: 1

Charles R Greathouse IV, Nov 13 2014

nonn

It appears that a(n) exists for each n, and that the sequence is increasing.

			a(3) = 7 because the Fibonacci-like sequence 2, 1, 3, 4, 7, 11, ... contains 7 and the sum of the first two terms is 3, while no smaller sequences work. (All terms must be nonnegative.)

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Index to sequences related to the complexity of n

Cf. A249783, A054495.

A249783[n_] := A249783[n] = Module[{a, k, A, B}, If[n<2, Return[n]]; For[k = 1, k <= n-1, k++, For[a=0, a <= k-1, a++, A = a; B = k-A; While[BA249783[k] == n, Return[k]]]; Array[a, 50] (* Jean-François Alcover, Jan 06 2017, adapted from PARI *)

A054495(n)=fordiv(n,d,if(A010056(n/d),return(d)))
A249783(n)=if(n<2,return(n));for(k=1,n-1,for(a=0,k-1,my(A=a,B=k-A);while(B=least&&t<=#v&&v[t]==0,v[t]=n;while(v[least],if(least++>#v,return(v)))))

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A054494 Largest Fibonacci factor of n.

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A249783 Smallest index of a Fibonacci-like sequence containing n, see comments.

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A280695 a(n) = n / A280694(n); n divided by the largest Lucas number (A000032) dividing n.

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A246399 Least k such that A249783(k) = n.

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