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 1-3 of 3 results.

A349100 a(n) is the product of the new Fibonacci divisors that appear when A129655(n) sets a new record for number of Fibonacci divisors.

Original entry on oeis.org

1, 2, 3, 8, 5, 144, 21, 55, 13, 34, 2584, 377, 6765, 46368
Offset: 1

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Author

Bernard Schott, Jul 16 2022

Keywords

Comments

As A129655(n) is also, up to A129655(14), the smallest integer that has exactly n Fibonacci divisors (A000045), a(n) from 1..14 is the new Fibonacci divisor that appears.
Kevin Ryde remarks that for two of the conjectured later terms of A129655, there are more than a single new Fibonacci divisor.

Examples

			A129655(1) = 1 because the smallest integer that has only one Fibonacci divisor is 1; the corresponding Fibonacci divisor is 1, so a(1) = 1.
A129655(6) = 720 and the set of the six Fibonacci divisors of 720 is {1, 2, 3, 5, 8, 144}. Then, A129655(7) = 5040 and the set of the seven Fibonacci divisors of 5040 is {1, 2, 3, 5, 8, 21, 144}. The new Fibonacci divisor that appears in this set is 21, hence a(7) = 21.
A129655(7) = 5040 and the set of the seven Fibonacci divisors of 5040 is {1, 2, 3, 5, 8, 21, 144}. Then A129655(8) = 55440 and the set of the eight Fibonacci divisors of 55040 is {1, 2, 3, 5, 8, 21, 55, 144}. The new Fibonacci divisor that appears is 55, hence a(8) = 55.

Crossrefs

Cf. A000045, A005086, A129655.

A005086 Number of Fibonacci numbers 1,2,3,5,... dividing n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 2, 1, 4, 1, 3, 2, 3, 2, 3, 1, 2, 3, 4, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 3, 3, 2, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 2, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 4, 1, 4, 4
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Indices of records are in A129655. - R. J. Mathar, Nov 02 2007

Links

Crossrefs

Cf. A000045, A038663, A051731, A010056, A072649, A079586, A129655.

Programs

  • Maple
    with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k))-floor((n-1)/fibonacci(k)), k=2..15)) od:
  • Mathematica
    f[n_] := Block[{k = 1}, While[Fibonacci[k] <= n, k++ ]; Count[ Mod[n, Array[ Fibonacci, k - 1]], 0] - 1]; Array[f, 105] (* Robert G. Wilson v, Dec 10 2006 *)
  • PARI
    isfib(n)=my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))
a(n)=sumdiv(n,d,isfib(d)) \\ Charles R Greathouse IV, Nov 06 2014
  • Python
    from sympy import divisors
from sympy.ntheory.primetest import is_square
def A005086(n): return sum(1 for d in divisors(n,generator=True) if is_square(m:=5*d**2-4) or is_square(m+8)) # Chai Wah Wu, Mar 30 2023
  • Python
    from itertools import count, takewhile
def F(f=1, g=1):
    while True:
        f, g = g, f+g;
        yield f
def a(n):
    return sum(1 for f in takewhile(lambda x: x<=n, F()) if n%f == 0)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Apr 03 2023

Formula

a(n) <= A072649(n). - Robert G. Wilson v, Dec 10 2006
Equals A051731 * A010056. - Gary W. Adamson, Nov 06 2007
G.f.: Sum_{n>=2} x^F(n)/(1-x^F(n)) where F(n) = A000045(n). - Joerg Arndt, Jan 06 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - Amiram Eldar, Dec 31 2023

Extensions

More terms from James Sellers, Feb 19 2001
Incorrect comment removed by Charles R Greathouse IV, Nov 06 2014

A356062 a(n) is the smallest integer that has exactly n Lucas divisors (A000032).

Original entry on oeis.org

1, 2, 4, 12, 36, 252, 2772, 52668, 1211364, 35129556, 1089016236, 44649665676, 2098534286772, 417608323067628, 88115356167269508, 24760415083002731748, 7948093241643876891108, 4140956578896459860267268
Offset: 1

Views

Author

Bernard Schott, Jul 25 2022

Keywords

Comments

The new Lucas numbers that appear at each step are in A356063.

Examples

			36 is divisible by 1, 2, 3, 4, 18, which are all Lucas numbers, and no integer < 36 has 5 divisors that are Lucas numbers, hence a(5) = 36.

Crossrefs

Cf. A000032, A304092, A356063.
Similar sequences: A087997 (palindromes), A129655 (Fibonacci), A333456 (Niven).

Extensions

a(10) from Amiram Eldar, Jul 25 2022
a(11)-a(18) from David A. Corneth, Jul 27 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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