A105800 -id:A105800 - cp's OEIS frontend

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

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).

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)).

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

a(A001605(n)) = 2; a(A105802(n)) = n.

It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */

{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Corrected and extended by Sascha Kurz, Jan 26 2003

Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.

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

Original entry on oeis.org

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)).

A280689 a(n) = A000045(A032742(n)) / A000045(A054576(n)), where A000045(n) gives the n-th Fibonacci number, A032742(n) = the largest proper divisor of n, and A054576(n) = A032742(A032742(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 5, 1, 4, 1, 13, 5, 7, 1, 17, 1, 11, 13, 89, 1, 18, 5, 233, 17, 29, 1, 122, 1, 47, 89, 1597, 13, 76, 1, 4181, 233, 123, 1, 842, 1, 199, 122, 28657, 1, 322, 13, 15005, 1597, 521, 1, 5777, 89, 843, 4181, 514229, 1, 1364, 1, 1346269, 842, 2207, 233, 39602, 1, 3571, 28657, 709805, 1, 5778, 1, 24157817, 15005, 9349, 89, 271442, 1, 15127, 5777

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

A000045 is a divisibility sequence, which guarantees that the result of the division is an integer.

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

Cf. A000045, A032742, A054576, A105800, A280688, A280690, A280691.

(define (A280689 n) (/ (A105800 n) (A280688 n)))

a(n) = A105800(n) / A280688(n) = A105800(n) / A105800(A032742(n)).
a(n) = A000045(A032742(n)) / A000045(A054576(n)).
a(n) = A280690(A032742(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)).

A280698 Greatest Lucas number that is a proper divisor of the n-th Lucas number, a(1) = 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 4, 3, 1, 7, 1, 3, 11, 1, 1, 18, 1, 7, 29, 3, 1, 47, 11, 3, 76, 7, 1, 123, 1, 1, 199, 3, 29, 322, 1, 3, 521, 47, 1, 843, 1, 7, 1364, 3, 1, 2207, 29, 123, 3571, 7, 1, 5778, 199, 47, 9349, 3, 1, 15127, 1, 3, 24476, 1, 521, 39603, 1, 7, 64079, 843, 1, 103682, 1, 3, 167761, 7, 199, 271443, 1, 2207, 439204, 3, 1, 710647, 3571, 3, 1149851, 47, 1

Offset: 1

Antti Karttunen, Jan 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..521

Cf. A000032, A000204, A105800, A280696, A280699.

(define (A280698 n) (A280696 (A000204 n)))

a(n) = A280696(A000204(n)).

cp's OEIS Frontend

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

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

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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

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

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A280689 a(n) = A000045(A032742(n)) / A000045(A054576(n)), where A000045(n) gives the n-th Fibonacci number, A032742(n) = the largest proper divisor of n, and A054576(n) = A032742(A032742(n)).

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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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A280698 Greatest Lucas number that is a proper divisor of the n-th Lucas number, a(1) = 1.

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