A005086 -id:A005086 - cp's OEIS frontend

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

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

Henry Bottomley, Apr 04 2000

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A010056, A005086, A037943.

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

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

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

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

A293431 a(n) is the number of Jacobsthal numbers dividing n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 3, 2, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 15, whose divisors are [1, 3, 5, 15], the first three, 1, 3 and 5 are all in A001045, thus a(15) = 3.
For n = 21, whose divisors are [1, 3, 7, 21], 1, 3 and 21 are in A001045, thus a(21) = 3.
For n = 105, whose divisors are [1, 3, 5, 7, 15, 21, 35, 105], only the divisors 1, 3, 5 and 21 are in A001045, thus a(105) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..21845

Cf. A000005, A001045, A147612, A293432, A293433.

Cf. also A005086.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, 1 &, MemberQ[s, #] &] &, 105]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293431(n) = sumdiv(n,d,A147612(d));

from sympy import divisors
def A293431(n): return sum(1 for d in divisors(n,generator=True) if (m:=3*d+1).bit_length()>(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

a(n) = Sum_{d|n} A147612(d).
a(n) = A293433(n) + A147612(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A001045(n) = 1.718591611927... . - Amiram Eldar, Jan 01 2024

A129655 Numbers that set a new record for number of Fibonacci divisors.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 55440, 720720, 12252240, 232792560, 6750984240, 276790353840, 12732356276640, 523410559111440, 24076885719126240, 1131613628798933280, 100713612963105061920, 20042008979657907322080

Offset: 1

Jason Earls, May 19 2007

From Donovan Johnson, Jul 07 2009: (Start)
a(15) <= 598420745002080,
a(16) <= 36503665445126880,
a(17) <= 1131613628798933280,
a(18) <= 100713612963105061920. (End)
From Robert Israel, Sep 26 2019: (Start)
a(15) <= 523410559111440,
a(16) <= 24076885719126240. (End)
From David A. Corneth, Sep 27 2019: (Start)
a(19) <= 20042008979657907322080,
a(20) <= 4669788092260292406044640,
a(21) <= 1312210453925142166098543840,
a(22) <= 414821946023574034721351415840,
a(23) <= 116564966832624303756699747851040,
a(24) <= 37417354353272401505900619060183840,
a(25) <= 19494441618054921184574222530355780640,
a(26) <= 31132623264033709131765033380978181682080,
a(27) <= 67277598873576845433744237136293850614974880. (End)
From a(1) up to a(14), last known term, this sequence is equivalent to: a(n) is the smallest number that has exactly n Fibonacci divisors (A000045). The products of the new Fibonacci divisors that appear successively are in A349100. - Bernard Schott, Jul 15 2022

			5040 has 60 divisors with 7 of them being Fibonacci numbers, namely 1, 2, 3, 5, 8, 21 and 144.

J. Earls, Red Zen, Lulu Press, NY, 2006, p. 105.

David A. Corneth, Some actual values and some upper bounds on a(n), for n=1..134
David A. Corneth, PARI program

Cf. A000045, A005086, A035105, A349100.

a(n) <= A035105(n+1). - Daniel Suteu, Sep 27 2019

More terms from Donovan Johnson, Feb 26 2008
a(14) from Donovan Johnson, Jul 07 2009
a(15)-a(19) confirmed by David A. Corneth, Sep 06 2024

A239930 Number of distinct quarter-squares dividing n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 19 2014

nonn

For more information about the quarter-squares see A002620.

			For n = 12 the quarter-squares <= 12 are [0, 0, 1, 2, 4, 6, 9, 12]. There are five quarter-squares that divide 12; they are [1, 2, 4, 6, 12], so a(12) = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Table of divisors.

Cf. A000005, A001221, A001511, A002620, A005086, A006519, A007862, A013661, A027750, A046951, A147645, A236103.

Cf. A240025.

a239930 = sum . map a240025 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

isA002620 := proc(n)
    local k,qsq ;
    for k from 0 do
        qsq := floor(k^2/4) ;
        if n = qsq then
            return true;
        elif qsq > n then
            return false;
        end if;
    end do:
end proc:
A239930 := proc(n)
    local a,d ;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if isA002620(d) then
            a:= a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 03 2014

qsQ[n_] := AnyTrue[Range[Ceiling[2 Sqrt[n]]], n == Floor[#^2/4]&]; a[n_] := DivisorSum[n, Boole[qsQ[#]]&]; Array[a, 110] (* Jean-François Alcover, Feb 12 2018 *)

a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ Amiram Eldar, Dec 31 2023

a(n) = Sum_{k=1..A000005(n)} A240025(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) + 1 = A013661 + 1 = 2.644934... . - Amiram Eldar, Dec 31 2023

A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 4, 1, 1, 2, 2, 1, 3, 1, 3, 2, 1, 2, 3, 2, 1, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 1, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 3, 3, 2, 1, 2, 2, 1, 1, 4, 1, 1, 3, 3, 1, 3, 2, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000204, A093540, A304092, A304093, A304096.

Cf. also A005086, A300837.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304094(n) = sumdiv(n,d,isA000204(d));

a(n) = A304092(n) - A059841(n).
a(n) = A304096(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 = 1.962858... . - Amiram Eldar, Dec 31 2023

A294880 Number of divisors of n that are in Perrin sequence, A001608.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 0, 3, 0, 2, 2, 1, 1, 2, 0, 3, 2, 2, 0, 3, 1, 1, 1, 2, 1, 4, 0, 1, 1, 2, 2, 3, 0, 1, 2, 3, 0, 3, 0, 2, 2, 1, 0, 3, 1, 3, 3, 1, 0, 2, 1, 2, 1, 2, 0, 5, 0, 1, 2, 1, 1, 3, 0, 3, 1, 4, 0, 3, 0, 1, 2, 1, 1, 3, 0, 3, 1, 1, 0, 4, 2, 1, 2, 2, 0, 5, 1, 1, 1, 1, 1, 3, 0, 2, 1, 3, 0, 4, 0, 1, 3

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n = 22, with divisors [1, 2, 11, 22], both 2 and 22 are in A001608, thus a(22) = 2.
For n = 644, with divisors [1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644], 2, 7 and 644 are in A001608, thus a(644) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..24914

Cf. A001608, A014981, A074788, A294878, A294879.

Cf. also A005086, A293431.

With[{s = LinearRecurrence[{0, 1, 1}, {3, 2, 5}, 15]}, Table[DivisorSum[n, 1 &, MemberQ[s, #] &], {n, 1, s[[-1]]}]] (* Amiram Eldar, Jan 01 2024 *)

A001608(n) = if(n<0, 0, polsym(x^3-x-1, n)[n+1]);
A294878(n) = { my(k=1,v); while((v=A001608(k))A294880(n) = sumdiv(n,d,A294878(d));

a(n) = Sum_{d|n} A294878(d).
a(n) = A294879(n) + A294878(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -1/5 + Sum_{n>=3} 1/A001608(n) = 1.603595519775230150708... . - Amiram Eldar, Jan 01 2024

A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 62, 86, 119, 159, 211, 273, 352, 446, 562, 697, 864, 1054, 1284, 1550, 1860, 2220, 2639, 3114, 3669, 4293, 5011, 5823, 6745, 7783, 8956, 10268, 11747, 13390, 15237, 17281, 19561, 22089, 24889, 27979, 31405, 35157, 39309, 43856, 48849, 54319, 60309, 66840, 73992, 81760, 90243

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into Fibonacci parts (with a single type of 1).

Convolution of A003107 and A005086.

			a(5) = 18 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 2 + 3 + 3 + 4 + 5 = 18.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for related partition-counting sequences

Cf. A000045, A003107, A005086, A240225, A319394.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
       b(n, h(i-1))+(p->p+[0, p[1]])(b(n-i, h(min(n-i, i)))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 18 2018

Rest[CoefficientList[Series[Sum[x^Fibonacci[k]/(1 - x^Fibonacci[k]), {k, 2, 20}]/Product[1 - x^Fibonacci[k], {k, 2, 20}], {x, 0, 52}], x]]

G.f.: Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

a(n) = Sum_{k=1..n} k * A319394(n,k). - Alois P. Heinz, Sep 18 2018

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

Bernard Schott, Jul 16 2022

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.

			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.

Cf. A000045, A005086, A129655.

A244964 Number of distinct generalized pentagonal numbers dividing n.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 4, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 2, 2, 2, 3, 1, 2, 1, 6, 1, 3, 1, 2, 3, 2, 3, 3, 1, 4, 1, 2, 1, 4, 2, 2, 1, 3, 1, 4, 2, 3, 1, 2, 2, 3, 1, 3, 1, 4, 1, 3, 1, 3, 5

Offset: 1

Omar E. Pol, Jul 10 2014

nonn

For more information about the generalized pentagonal numbers see A001318.

			For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

From Amiram Eldar, Dec 31 2023: (Start)
a(n) = Sum_{d|n} A080995(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

A340542 Number of Fibonacci divisors of Fibonacci(n)^2 + 1.

Original entry on oeis.org

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

Offset: 0

Michel Lagneau, Jan 12 2021

nonn

A Fibonacci divisor of a number k is a Fibonacci number that divides k.
It is interesting to compare this sequence with A339669.
We observe that a(2n) = A339669(2n) if n = 5*k + 2 or n = 5*k + 3, with k >= 0, because Lucas(2n)^2 = 5*Fibonacci(2n)^2 + 4 (see A005248: all nonnegative integer solutions of the Pell equation a(n)^2 - 5*b(n)^2 = +4 together with b(n)= A001906(n), n>=0. - Wolfdieter Lang, Aug 31 2004).
So, Lucas(2n)^2 + 1 = 5*(Fibonacci(2n)^2 + 1). Lucas(2n)^2 + 1 and Fibonacci(2n)^2 + 1 have the same Fibonacci divisors for n = 5*k + 2 or n = 5*k + 3. For the other values of n = 5*k, 5*k + 1 or 5*k + 4, 5 is a Fibonacci divisor of Lucas(2n)^2 + 1 but not of Fibonacci(2n)^2 + 1. So for these last three values of n, a(2n) = A339669(2n) - 1 (except for m = 1 and 2, 5*F(m) is never a Fibonacci number).

			a(13) = 5 because the 5 Fibonacci divisors of Fibonacci(13)^2 + 1 = 233^2 + 1 are 1, 2, 5, 89 and 610.
a(16) = 5 because the 5 Fibonacci divisors of Fibonacci(16)^2 + 1 = 987^2 + 1 are 1, 2, 5, 610, and 1597.
Remark: the 5 Fibonacci divisors of Lucas(16)^2 + 1 = 2207^2 + 1 are 1, 2, 5, 610, and 1597, the index 16 = 2*8 with 8 of the form 5*k + 3.

Michel Marcus, Table of n, a(n) for n = 0..200

Cf. A000032, A000045, A005248, A010056, A339461, A339669.

Cf. A005086, A245306.

with(combinat,fibonacci):nn:=100:F:={}:
for k from 0 to nn do:
  F:=F union {fibonacci(k)}:
od:
   for m from 0 to 90 do:
    f:=fibonacci(m)^2+1:d:=numtheory[divisors](f):
    lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
   od:

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
a(n) = sumdiv(fibonacci(n)^2+1, d, isfib(d)); \\ Michel Marcus, Jan 12 2021

a(n) = A005086(A245306(n)). - Michel Marcus, Aug 10 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A293431 a(n) is the number of Jacobsthal numbers dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A129655 Numbers that set a new record for number of Fibonacci divisors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A239930 Number of distinct quarter-squares dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A294880 Number of divisors of n that are in Perrin sequence, A001608.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple