cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Antti Karttunen, Jan 11 2017

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A000045, A032742, A105800, A280687, A280696.
Cf. A001690 (gives the positions n > 1 where this sequence and A054494 obtain equal values).

Programs

  • PARI
    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
  • Scheme
    ;; 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)))))

Formula

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

A054495 Smallest k such that n/k is a Fibonacci number.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 7, 1, 3, 2, 11, 4, 1, 7, 3, 2, 17, 6, 19, 4, 1, 11, 23, 3, 5, 2, 9, 14, 29, 6, 31, 4, 11, 1, 7, 12, 37, 19, 3, 5, 41, 2, 43, 22, 9, 23, 47, 6, 49, 10, 17, 4, 53, 18, 1, 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
Offset: 1

Views

Author

Henry Bottomley, Apr 04 2000

Keywords

Examples

			a(10)=2 because 10/1=10 is not a Fibonacci number but 10/2=5 is.

Links

Crossrefs

Cf. A000045, A010056, A005086, A037943.

Programs

  • PARI
    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(d))) \\ Charles R Greathouse IV, Nov 05 2014
  • Python
    from sympy import divisors
from sympy.ntheory.primetest import is_square
def A054495(n): return next(d for d in divisors(n) if is_square(m:=5*(n//d)**2-4) or is_square(m+8)) # Chai Wah Wu, May 06 2024

Formula

a(n) = n/A054494(n). [Corrected by Charles R Greathouse IV, Nov 05 2014]

Extensions

a(34), a(55), a(68) corrected by Charles R Greathouse IV, Nov 06 2014

A280694 Largest Lucas number (A000032) dividing n.

Original entry on oeis.org

1, 2, 3, 4, 1, 3, 7, 4, 3, 2, 11, 4, 1, 7, 3, 4, 1, 18, 1, 4, 7, 11, 1, 4, 1, 2, 3, 7, 29, 3, 1, 4, 11, 2, 7, 18, 1, 2, 3, 4, 1, 7, 1, 11, 3, 2, 47, 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, 76, 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
Offset: 1

Views

Author

Antti Karttunen, Jan 11 2017

Keywords

Links

Crossrefs

Cf. A000032, A054494, A280695, A280699.
Cf. A057854 (gives the positions n > 1 where this sequence and A280696 obtain equal values).

Programs

  • Scheme
    ;; A stand-alone program:
(define (A280694 n) (let loop ((l1 1) (l2 3) (lpd 1)) (cond ((> l1 n) (if (and (= 1 lpd) (even? n)) 2 lpd)) ((zero? (modulo n l1)) (loop l2 (+ l1 l2) l1)) (else (loop l2 (+ l1 l2) lpd)))))

Formula

a(n) = n / A280695(n).
Other identities. For all n >= 1:
a(A000032(n)) = A000032(n).
a(A057854(n)) = A280696(A057854(n)).
a(A000045(n)) = A280699(n).

A372048 The index of the largest Fibonacci number that divides the sum of Fibonacci numbers with indices 1 through n.

Original entry on oeis.org

2, 3, 3, 2, 4, 5, 4, 4, 6, 7, 6, 6, 8, 9, 8, 8, 10, 11, 10, 10, 12, 13, 12, 12, 14, 15, 14, 14, 16, 17, 16, 16, 18, 19, 18, 18, 20, 21, 20, 20, 22, 23, 22, 22, 24, 25, 24, 24, 26, 27, 26, 26, 28, 29, 28, 28, 30, 31, 30, 30, 32, 33, 32, 32, 34, 35, 34, 34, 36, 37, 36, 36, 38, 39, 38, 38, 40, 41, 40, 40
Offset: 1

Views

Author

Tanya Khovanova and the MIT PRIMES STEP senior group, Apr 17 2024

Keywords

Comments

The sum of the first n Fibonacci numbers is sequence A000071.
When we divide the sum by the largest Fibonacci number that divides the sum, we always get a Lucas number.
For n > 3, a(n+4) = a(n) + 2.

Examples

			The sum of the first three Fibonacci numbers is 1+1+2=4. The largest Fibonacci that divides this sum is 2, the third Fibonacci number. Thus, a(3) = 2. After the division, we get 4/2 = 2, the zeroth Lucas number.
The sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. Thus, a(10) = 7. After the division, we get 143/13 = 11, the fifth Lucas number.

Links

Crossrefs

Cf. A000032, A000045, A000071, A054494, A075850, A372049.

Programs

Formula

G.f.: x*(x^6-2*x^5+2*x^4-2*x^3+x^2-x+2)/((x^2+1)*(x-1)^2). - Alois P. Heinz, Jul 25 2025

A372050 The index of the largest Fibonacci number that divides the sum of Fibonacci numbers with indices 1 through A000217(n) (the n-th triangular number).

Original entry on oeis.org

2, 3, 5, 7, 8, 12, 14, 18, 24, 28, 35, 41, 46, 54, 60, 68, 78, 89, 97, 107, 116, 128, 138, 150, 164, 176, 191, 205, 218, 234, 248, 264, 282, 298, 317, 335, 352, 372, 390, 410, 432, 452, 475, 497, 518, 542, 564, 588, 614, 638, 665, 691, 716, 744, 770, 798, 828, 856, 887, 917, 946, 978
Offset: 1

Views

Author

Tanya Khovanova and MIT PRIMES STEP senior group, Apr 17 2024

Keywords

Comments

When we divide the sum by the largest Fibonacci number that divides the sum, we always get a Lucas number.

Examples

			For example, the sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. Thus, as 10 is the fourth triangular number a(4) = 7. After the division we get 143/13 = 11, the fifth Lucas number.

Crossrefs

Cf. A000032, A000045, A000071, A000217, A054494, A075850, A372048, A372049, A372051.

A355758 Irregular triangle read by rows in which row n lists the divisors of n that are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 3, 1, 1, 2, 8, 1, 3, 1, 2, 5, 1, 1, 2, 3, 1, 13, 1, 2, 1, 3, 5, 1, 2, 8, 1, 1, 2, 3, 1, 1, 2, 5, 1, 3, 21, 1, 2, 1, 1, 2, 3, 8, 1, 5, 1, 2, 13, 1, 3, 1, 2, 1, 1, 2, 3, 5, 1, 1, 2, 8, 1, 3, 1, 2, 34, 1, 5, 1, 2, 3, 1, 1, 2, 1, 3, 13, 1, 2, 5, 8
Offset: 1

Views

Author

Michel Marcus, Jul 16 2022

Keywords

Examples

			Irregular triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2;
  1, 5;
  1, 2, 3;
  1;
  1, 2, 8;
  1, 3;
  1, 2, 5;
  ...

Crossrefs

Cf. A000045, A010056.
Cf. A000012 (left border), A054494 (right border).
Cf. A005086 (row lengths), A005092 (row sums).
Subsequence of A027750.

Programs

  • Mathematica
    With[{fib = Fibonacci[Range[2, 10]]}, row[n_] := Select[Divisors[n], MemberQ[fib, #] &]; Table[row[n], {n, 1, fib[[-1]]}] // Flatten] (* Amiram Eldar, Jul 16 2022 *)
  • PARI
    isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
row(n) = select(isfib, divisors(n));
Showing 1-6 of 6 results.

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