A280694 -id:A280694 - cp's OEIS frontend

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

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)))
```

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).

A280699 Greatest Lucas number that is a divisor of the n-th Fibonacci number, a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 1, 7, 2, 11, 1, 18, 1, 29, 2, 47, 1, 76, 1, 123, 2, 199, 1, 322, 1, 521, 2, 843, 1, 1364, 1, 2207, 2, 3571, 1, 5778, 1, 9349, 2, 15127, 1, 24476, 1, 39603, 2, 64079, 1, 103682, 1, 167761, 2, 271443, 1, 439204, 1, 710647, 2, 1149851, 1, 1860498, 1, 3010349, 2, 4870847, 1, 7881196, 1, 12752043, 2, 20633239, 1, 33385282, 1, 54018521, 2, 87403803

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

The even bisection is almost certainly A000204. Consider for example the well-known formula L(n)*F(n) = F(2n) = A001906(n).

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

Cf. A000045, A000032, A000204, A001906, A105800, A280694, A280698.

(define (A280699 n) (A280694 (A000045 n)))

a(n) = A280694(A000045(n)).

cp's OEIS Frontend

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

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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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A280699 Greatest Lucas number that is a divisor of the n-th Fibonacci number, a(1) = a(2) = 1.

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