A280686 -id:A280686 - cp's OEIS frontend

A280687 a(n) = n / A280686(n); n divided by its largest Fibonacci proper divisor, a(1) = 1.

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 4, 13, 7, 3, 2, 17, 6, 19, 4, 7, 11, 23, 3, 5, 2, 9, 14, 29, 6, 31, 4, 11, 17, 7, 12, 37, 19, 3, 5, 41, 2, 43, 22, 9, 23, 47, 6, 49, 10, 17, 4, 53, 18, 11, 7, 19, 29, 59, 12, 61, 31, 3, 8, 5, 22, 67, 2, 23, 14, 71, 9, 73, 37, 15, 38, 77, 6, 79, 10, 27, 41, 83, 4, 17, 43, 29, 11, 89, 18, 7, 46, 31, 47, 19, 12, 97, 49, 33, 20

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

For n > 1, a(n) = n divided by the greatest Fibonacci number that divides n and is less than n.

Antti Karttunen, Table of n, a(n) for n = 1..10946

Cf. A000045, A280686, A280690, A280697.

Scheme
```
(define (A280687 n) (/ n (A280686 n)))
```

a(n) = n / A280686(n).

a(A000045(n)) = A280690(n).

A280688 a(n) = A000045(A054576(n))= A280686(A105800(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 5, 1, 1, 1, 8, 1, 1, 2, 13, 1, 5, 1, 21, 1, 1, 1, 34, 1, 1, 1, 55, 1, 13, 1, 89, 5, 1, 1, 144, 1, 5, 1, 233, 1, 34, 1, 377, 1, 1, 1, 610, 1, 1, 13, 987, 1, 89, 1, 1597, 1, 13, 1, 2584, 1, 1, 5, 4181, 1, 233, 1, 6765, 34, 1, 1, 10946, 1, 1, 1, 17711, 1, 610, 1, 28657, 1, 1, 1, 46368, 1, 13, 89, 75025, 1, 1597

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..987

Cf. A000045, A032742, A054576, A105800, A280686, A280689.

a(n) = A000045(A054576(n)).
a(n) = A280686(A105800(n)).
a(n) = A105800(A032742(n)).

A105800 Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 8, 1, 13, 5, 21, 1, 34, 1, 55, 13, 89, 1, 144, 5, 233, 34, 377, 1, 610, 1, 987, 89, 1597, 13, 2584, 1, 4181, 233, 6765, 1, 10946, 1, 17711, 610, 28657, 1, 46368, 13, 75025, 1597, 121393, 1, 196418, 89, 317811, 4181, 514229, 1, 832040, 1

Offset: 1

Reinhard Zumkeller, Apr 20 2005

nonn

a(A001605(n)) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..987
Eric Weisstein's World of Mathematics, Fibonacci Number
Wikipedia, Divisibility sequence

Cf. A000045, A032742, A076984, A139045, A280686, A280688, A280689, A280690, A280698, A280699.

Cf. A046022 (gives the positions of ones).

nn=70;Join[{1,1},With[{fibs=Fibonacci[Range[nn]]},Table[ Max[ Intersection[ Most[Divisors[fibs[[n]]]],fibs]],{n,3,nn}]]] (* Harvey P. Dale, Apr 10 2012 *)

From Antti Karttunen, Jan 11 2017: (Start)
a(n) = A280686(A000045(n)).
a(n) = A000045(A032742(n)). [Because A000045 is a divisibility sequence.]
a(A032742(n)) = A280688(n).
(End)

A054494 Largest Fibonacci factor of n.

Original entry on oeis.org

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

A280696 Largest Lucas proper divisor of n, a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 7, 3, 4, 1, 3, 1, 4, 7, 11, 1, 4, 1, 2, 3, 7, 1, 3, 1, 4, 11, 2, 7, 18, 1, 2, 3, 4, 1, 7, 1, 11, 3, 2, 1, 4, 7, 2, 3, 4, 1, 18, 11, 7, 3, 29, 1, 4, 1, 2, 7, 4, 1, 11, 1, 4, 3, 7, 1, 18, 1, 2, 3, 4, 11, 3, 1, 4, 3, 2, 1, 7, 1, 2, 29, 11, 1, 18, 7, 4, 3, 47, 1, 4, 1, 7, 11, 4, 1, 3, 1, 4, 7, 2, 1, 18, 1, 11, 3, 7, 1, 3, 1

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

For n > 1, a(n) = greatest Lucas number (A000032) that divides n and is less than n.

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

Cf. A000032, A000045, A000204, A280686, A280697, A280698, A280699.

Cf. A057854 (gives the positions n > 1 where this sequence and A280694 obtain equal values).

;; A stand-alone program:
(define (A280696 n) (let loop ((l1 1) (l2 3) (lpd 1)) (cond ((>= l1 n) (if (and (= 1 lpd) (even? n) (> n 2)) 2 lpd)) ((zero? (modulo n l1)) (loop l2 (+ l1 l2) l1)) (else (loop l2 (+ l1 l2) lpd)))))

a(n) = n / A280697(n).
Other identities. For all n >= 1:
a(A057854(n)) = A280694(A057854(n)).
a(A000204(n)) = A280698(n).

cp's OEIS Frontend

A280687 a(n) = n / A280686(n); n divided by its largest Fibonacci proper divisor, a(1) = 1.

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A280688 a(n) = A000045(A054576(n))= A280686(A105800(n)).

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A105800 Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.

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A054494 Largest Fibonacci factor of n.

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A280696 Largest Lucas proper divisor of n, a(1) = a(2) = 1.

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