A280687 -id:A280687 - cp's OEIS frontend

A280686 Largest Fibonacci proper divisor of n, a(1) = 1.

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

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

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

			For n=3, the greatest Fibonacci number that divides 3 and is less than 3 is A000045(1)=A000045(2)=1, thus a(3) = 1.
For n=20, the greatest Fibonacci number that divides 20 and is less than 20 is A000045(5)=5, thus a(20) = 5.
For n=21, the greatest Fibonacci number that divides 21 and is less than 21 is A000045(4)=3, thus a(21) = 3.

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

Cf. A000045, A032742, A105800, A280687, A280696.

Cf. A001690 (gives the positions n > 1 where this sequence and A054494 obtain equal values).

a(n)=my(r=1,lim=if(n%2,n\3,n/2),a=1,b=2); while(bCharles R Greathouse IV, Jun 20 2017

;; A stand-alone program:
(define (A280686 n) (let loop ((f1 1) (f2 1) (lpd 1)) (cond ((>= f2 n) lpd) ((zero? (modulo n f2)) (loop f2 (+ f1 f2) f2)) (else (loop f2 (+ f1 f2) lpd)))))

a(n) = n / A280687(n).
Other identities. For all n >= 1:
a(A000045(n)) = A105800(n).
a(A001690(n)) = A054494(A001690(n)).

A280690 a(n) = A000045(n) / A105800(n); the n-th Fibonacci number divided by its largest Fibonacci proper divisor.

Original entry on oeis.org

1, 1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 18, 233, 29, 122, 47, 1597, 76, 4181, 123, 842, 199, 28657, 322, 15005, 521, 5777, 843, 514229, 1364, 1346269, 2207, 39602, 3571, 709805, 5778, 24157817, 9349, 271442, 15127, 165580141, 24476, 433494437, 39603, 1860497, 64079, 2971215073, 103682, 598364773, 167761, 12752042, 271443

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

A000045 is a divisibility sequence.

Antti Karttunen, Table of n, a(n) for n = 1..377
Wikipedia, Divisibility sequence

Cf. A000045, A032742, A105800, A280687, A280689.

a(n) = A000045(n) / A105800(n) = A000045(n) / A000045(A032742(n)).
a(n) = A280687(A000045(n)).
Other identities. For all n >= 1:
a(A032742(n)) = A280689(n).

A280697 a(n) = n / A280696(n); n divided by its largest Lucas proper divisor, a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 5, 11, 3, 13, 2, 5, 4, 17, 6, 19, 5, 3, 2, 23, 6, 25, 13, 9, 4, 29, 10, 31, 8, 3, 17, 5, 2, 37, 19, 13, 10, 41, 6, 43, 4, 15, 23, 47, 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, 19, 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

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

For n > 1, a(n) = n divided by the 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, A280695, A280696, A280687.

Scheme
```
(define (A280697 n) (/ n (A280696 n)))
```

a(n) = n / A280696(n).

cp's OEIS Frontend

A280686 Largest Fibonacci proper divisor of n, a(1) = 1.

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A280690 a(n) = A000045(n) / A105800(n); the n-th Fibonacci number divided by its largest Fibonacci proper divisor.

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A280697 a(n) = n / A280696(n); n divided by its largest Lucas proper divisor, a(1) = 1.

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