A005086 -id:A005086 - cp's OEIS frontend

A300837 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.

1, 2, 2, 4, 2, 5, 3, 5, 4, 5, 3, 10, 2, 6, 5, 7, 4, 9, 4, 10, 5, 6, 3, 13, 5, 5, 7, 11, 3, 13, 4, 10, 8, 6, 6, 16, 3, 8, 5, 14, 4, 12, 4, 11, 10, 8, 3, 18, 6, 11, 9, 10, 5, 16, 5, 14, 7, 6, 4, 23, 4, 8, 9, 13, 6, 16, 5, 10, 7, 14, 4, 23, 4, 8, 12, 12, 8, 13, 4, 20, 10, 9, 5, 23, 9, 9, 8, 17, 2, 22, 6, 12, 8, 6, 8, 24, 3, 12, 13, 19, 5, 15, 4, 14, 13

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

			For n=12, its divisors are 1, 2, 3, 4, 6 and 12. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101, 1001 and 10101. Total number of 1's present is 10 (ten), thus a(12) = 10.

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

Cf. A000045, A007895, A014417, A072649, A300835, A300836.

Cf. also A005086, A093653.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); };
A300837(n) = sumdiv(n,d,A007895(d));

a(n) = Sum_{d|n} A007895(d).
a(n) = A300836(n) + A007895(n).
For all n >=1, a(n) >= A005086(n).

A005092 Sum of Fibonacci numbers 1,2,3,5,... that divide n.

Original entry on oeis.org

1, 3, 4, 3, 6, 6, 1, 11, 4, 8, 1, 6, 14, 3, 9, 11, 1, 6, 1, 8, 25, 3, 1, 14, 6, 16, 4, 3, 1, 11, 1, 11, 4, 37, 6, 6, 1, 3, 17, 16, 1, 27, 1, 3, 9, 3, 1, 14, 1, 8, 4, 16, 1, 6, 61, 11, 4, 3, 1, 11, 1, 3, 25, 11, 19, 6, 1, 37, 4, 8, 1, 14, 1, 3, 9, 3, 1, 19, 1

Offset: 1

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000045, A005086.

a:= n-> add(`if`(issqr(5*d^2+4) or issqr(5*d^2-4), d, 0)
        , d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 07 2017

nmax = 100; With[{fibs = Fibonacci[Range[2, Floor[Log[nmax*Sqrt[5]] / Log[GoldenRatio]] + 1]]}, Table[Total[Select[fibs, Divisible[n, #1] & ]], {n, 1, nmax}]] (* Vaclav Kotesovec, Apr 29 2019 *)

G.f.: Sum_{k>=2} F(k)*x^F(k)/(1 - x^F(k)), where F(k) is the k-th Fibonacci number (A000045). - Ilya Gutkovskiy, Jan 06 2017

A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 55, its proper divisors are [1, 5, 11], of which only two, namely 1 and 5 are in A000045, thus a(55) = 2.

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

Cf. A000045, A005086, A010056, A079586, A293436.

Cf. also A293433.

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ This function from Charles R Greathouse IV, Jul 30 2012
A293435(n) = sumdiv(n,d,(dA010056(d));

a(n) = Sum_{d|n, dA010056(d).
a(n) = A005086(n) - A010056(n).
G.f.: Sum_{k>=2} x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - Amiram Eldar, Jul 05 2025

A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A000032, A093540, A102460, A304091, A304094, A304096.

Cf. also A005086, A300837.

Module[{nn=11,luc},luc=LucasL[Range[0,nn]];Table[Count[n/luc,?IntegerQ],{n,Max[luc]}]] (* _Harvey P. Dale, Jul 01 2023 *)

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304092(n) = sumdiv(n,d,A102460(d));

a(n) = Sum_{d|n} A102460(d).
a(n) = A304091(n) + A102460(n).
a(n) = A304094(n) + A059841(n) = A304096(n) + A059841(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 + 1/2 = 2.462858... . - Amiram Eldar, Dec 31 2023

A054494 Largest Fibonacci factor of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 1, 8, 3, 5, 1, 3, 13, 2, 5, 8, 1, 3, 1, 5, 21, 2, 1, 8, 5, 13, 3, 2, 1, 5, 1, 8, 3, 34, 5, 3, 1, 2, 13, 8, 1, 21, 1, 2, 5, 2, 1, 8, 1, 5, 3, 13, 1, 3, 55, 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, 89, 5, 13, 2, 3, 2, 5, 8, 1, 2, 3, 5

Offset: 1

Henry Bottomley, Apr 04 2000

			a(10)=5 because 1, 2 and 5 are the Fibonacci numbers which divide 10 and 5 is the largest.

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

Cf. A000045, A005086, A010056, A054495.
Sequences with similar definitions: A047930 (smallest Fibonacci multiple), A280686 (restricted to proper divisors), A280694 (equivalent for Lucas numbers).
Positions of 1's: A147956.

With[{fibs=Fibonacci[Range[20]]},Table[Max[Select[fibs,Divisible[ n,#]&]],{n,100}]] (* Harvey P. Dale, Jul 17 2012 *)

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(n/d))) \\ Charles R Greathouse IV, Nov 05 2014

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A054494(n): return next(d for d in sorted(divisors(n,generator=True),reverse=True) if is_square(m:=5*d**2-4) or is_square(m+8)) # Chai Wah Wu, May 06 2024

a(n) = n/A054495(n).

Corrected by Harvey P. Dale, Jul 17 2012

A304105 Restricted growth sequence transform of A304104, a filter sequence related to how the divisors of n are expressed in Fibonacci number system.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 4, 5, 8, 2, 9, 4, 10, 11, 12, 11, 8, 6, 9, 13, 14, 15, 4, 16, 17, 5, 18, 11, 8, 19, 20, 9, 21, 13, 22, 4, 23, 11, 24, 25, 26, 27, 28, 5, 29, 30, 31, 32, 8, 33, 34, 6, 35, 36, 9, 11, 37, 25, 22, 12, 38, 39, 40, 33, 41, 16, 42, 25, 43, 11, 44, 45, 46, 47, 18, 11, 48, 49, 50, 51, 52, 53, 54, 19, 55, 2, 56, 9, 57, 22, 9, 58, 59, 13, 60

Offset: 1

Antti Karttunen, May 13 2018

nonn

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

Cf. A005086, A304096, A300837, A304101, A304103, A304104.

\\ Needs also code from A304101:
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304104(n))),"b304105.txt");

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A005086, A304096 or A300837} for example.

A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 13, 16, 18, 21, 22, 25, 27, 29, 32, 35, 36, 39, 40, 43, 46, 48, 49, 53, 55, 58, 60, 62, 63, 67, 68, 71, 73, 76, 78, 81, 82, 84, 87, 91, 92, 96, 97, 99, 102, 104, 105, 109, 110, 113, 115, 118, 119, 122, 125, 128, 130, 132, 133, 137, 138, 140, 143, 146

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

			a(15)=[ 15/1 ]+[ 15/2 ]+[ 15/3 ]+[ 15/5 ]+[ 15/8 ]+[ 15/13 ]+[ 15/21 ]+...=32.

Vaclav Kotesovec, Plot of a(n)/n for n = 1..100000

Cf. A005086.

[&+[Floor(n/Fibonacci(k+2)):k in [0..n]]:n in [1..64]]; // Marius A. Burtea, Jul 16 2019

with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k)), k=2..15)) od:

Table[Sum[Floor[n/Fibonacci[k] ],{k,2,200}],{n,70}] (* Harvey P. Dale, Jul 21 2021 *)
Table[Sum[Floor[n/Fibonacci[k]], {k, 2, Log[Sqrt[5]*n]/Log[GoldenRatio] + 1}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 30 2021 *)

G.f.: (1/(1 - x)) * Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Jul 16 2019

Conjecture: a(n) ~ c * n, where c = A079586 - 1. - Vaclav Kotesovec, Aug 30 2021

More terms from Simon Plouffe, who points out that the first differences give A005086

More terms from James Sellers, Feb 19 2001

A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 1

Antti Karttunen, May 13 2018

nonn

a(n) is the number of the divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

			The divisors of 4 are 1, 2 and 4. Of these only 4 is a Lucas number larger than 3, thus a(4) = 1.
The divisors of 28 are 1, 2, 4, 7, 14 and 28. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.

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

Cf. A000032, A000045, A000204, A093540, A102460, A304092, A304094, A304095, A304104.

Cf. also A005086, A300837.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304096(n) = sumdiv(n,d,(d>3)*A102460(d));

a(n) = Sum_{d|n, d>3} A102460(d).
a(n) = A304094(n) - A079978(n) - 1.
a(n) = A304092(n) - A059841(n) - A079978(n) - 1.
a(n) = A007949(A304104(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Dec 31 2023

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 20, 3, 12, 10, 20, 3, 420, 2, 30, 20, 60, 11, 300, 11, 420, 12, 30, 5, 4200, 22, 20, 130, 990, 3, 11000, 11, 420, 102, 44, 30, 31500, 5, 242, 20, 10920, 11, 3000, 13, 1170, 1100, 190, 3, 231000, 33, 2420, 506, 420, 19, 66300, 12, 9900, 110, 30, 11, 8085000, 13, 242, 300, 5460, 52, 56100, 19, 660, 130, 19500, 13, 9135000, 11, 290, 4180, 2178, 99

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304102, A304105 (restricted growth sequence transform of this sequence).

\\ Needs also code from A304101:
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };

a(n) = Product_{d|n, d>1} A000040(A304101(d)-1).
a(n) = (1/2) * A304102(n) * A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A001511(a(n)) = A005086(n).
A007949(a(n)) = A304096(n).

A339461 Number of Fibonacci divisors of n^2 + 1.

Original entry on oeis.org

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

Offset: 0

Michel Lagneau, Dec 06 2020

nonn

			a(13) = 4 because the divisors of 13^2 + 1 = 170 are {1, 2, 5, 10, 17, 34, 85, 170} with 4 Fibonacci divisors: 1, 2, 5 and 34.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A000045, A002522, A005086, A005478, A005574, A128428, A338762, A338794.

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

Array[DivisorSum[#^2 + 1, 1 &, Or @@ Map[IntegerQ@ Sqrt[#] &, 5 #^2 + 4 {-1, 1}] &] &, 105, 0] (* Michael De Vlieger, Dec 07 2020 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
a(n) = sumdiv(n^2+1, d, isfib(d)); \\ Michel Marcus, Dec 06 2020

a(A005574(n)) = 1 for n > 2.

a(n) = A005086(A002522(n)). - Michel Marcus, Dec 06 2020

cp's OEIS Frontend

A300837 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A005092 Sum of Fibonacci numbers 1,2,3,5,... that divide n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A304092 Number of Lucas numbers (A000032: 2, 1, 3, 4, 7, 11, ...) dividing n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A054494 Largest Fibonacci factor of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A304105 Restricted growth sequence transform of A304104, a filter sequence related to how the divisors of n are expressed in Fibonacci number system.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

Views

Author