A005092 -id:A005092 - cp's OEIS frontend

A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).

1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 41, 49, 59, 71, 83, 99, 115, 134, 157, 180, 208, 239, 272, 312, 353, 400, 453, 509, 573, 642, 717, 803, 892, 993, 1102, 1219, 1350, 1489, 1640, 1808, 1983, 2178, 2386, 2609, 2854, 3113, 3393, 3697, 4017, 4367, 4737

Offset: 0

N. J. A. Sloane, Herman P. Robinson

The partitions allow repeated items but the order of items is immaterial (1+2=2+1). - Ron Knott, Oct 22 2003

A098641(n) = a(A000045(n)). - Reinhard Zumkeller, Apr 24 2005

			a(4) = 4 since the 4 partitions of 4 using only Fibonacci numbers, repetitions allowed, are 1+1+1+1, 2+2, 2+1+1, 3+1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
G. Almkvist, Partitions with Parts in a Finite Set and with Parts Outside a Finite Set, Exper. Math. vol 11 no 4 (2002) p 449-456.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

Cf. A007000, A005092, A028290 (where the only Fibonacci numbers allowed are 1, 2, 3, 5 and 8).
Cf. A000045, A000119, A102848, A238998.
Row sums of A319394.

import Data.MemoCombinators (memo2, integral)
a003107 n = a003107_list !! n
a003107_list = map (p' 2) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m | m < fib   = 0
         | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
-- Reinhard Zumkeller, Dec 09 2015

F:= combinat[fibonacci]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

CoefficientList[ Series[1/ Product[1 - x^Fibonacci[i], {i, 2, 21}], {x, 0, 53}], x] (* Robert G. Wilson v, Mar 28 2006 *)
nmax = 53;
s = Table[Fibonacci[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
F = Fibonacci;
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0,
     b[n, i - 1] + If[F[i] > n, 0, b[n - F[i], i]]]];
a[n_] := Module[{j}, For[j = Floor@Log[(1+Sqrt[5])/2, n+1],
     F[j + 1] <= n, j++]; b[n, j]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

a(n) = (1/n)*Sum_{k=1..n} A005092(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Jan 21 2002
G.f.: Product_{i>=2} 1/(1-x^fibonacci(i)). - Ron Knott, Oct 22 2003
a(n) = f(n,1,1) with f(x,y,z) = if xReinhard Zumkeller, Nov 11 2009
G.f.: 1 + Sum_{i>=2} x^Fibonacci(i) / Product_{j=2..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

More terms from Vladeta Jovovic, Jan 21 2002

A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 35, 49, 66, 88, 115, 148, 189, 238, 297, 368, 451, 550, 665, 799, 956, 1136, 1344, 1583, 1855, 2167, 2520, 2920, 3373, 3882, 4455, 5097, 5814, 6617, 7509, 8502, 9604, 10823, 12173, 13662, 15302, 17110, 19093, 21271, 23657, 26266

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

			a(2)=4 because we have [2],[1',1'],[1',1],[1,1] (the two types of 1 are denoted 1 and 1').

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
James Propp and N. J. A. Sloane, Email, March 1994.

Cf. A003107.

Cf. A000045.

import Data.MemoCombinators (memo2, integral)
a007000 n = a007000_list !! n
a007000_list = map (p' 1) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m | m < fib   = 0
         | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
-- Reinhard Zumkeller, Dec 09 2015

with(combinat): gf := 1/product((1-q^fibonacci(k)), k=1..20): s := series(gf, q, 200): for i from 0 to 199 do printf(`%d,`,coeff(s, q, i)) od: # James Sellers, Feb 08 2002

CoefficientList[ Series[ 1/Product[1 - x^Fibonacci[i], {i, 1, 15}], {x, 0, 50}], x]
nmax = 46; f = Table[Fibonacci[n], {n, nmax}];
Table[Length[IntegerPartitions[n, All, f]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) = 1/n*Sum_{k=1..n} (A005092(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Aug 22 2002
G.f.: 1/Product_{j>=1} (1-x^fibonacci(j)). - Emeric Deutsch, Mar 05 2006
G.f.: Sum_{i>=0} x^Fibonacci(i) / Product_{j=1..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

More terms from James Sellers, Feb 08 2002

A293432 Sum of Jacobsthal numbers that divide n.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 12, 4, 1, 1, 9, 1, 1, 4, 1, 6, 25, 12, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 15, 1, 6, 4, 1, 1, 4, 6, 1, 25, 44, 12, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 17, 1, 4, 1, 1, 9, 1, 1, 25, 1, 6, 15, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 12, 4, 1, 6, 4, 1, 1, 25, 91, 44, 4, 12, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 15, 6, 1, 4, 1, 1, 30

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

a(n) is the sum of the divisors of n that are Jacobsthal numbers (A001045).

			For n = 15, whose divisors are [1, 3, 5, 15], the first three, 1, 3 and 5 are all in A001045, thus a(15) = 1 + 3 + 5 = 9.
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) = 1 + 3 + 5 + 21 = 30.
For n = 21845, whose divisors are [1, 5, 17, 85, 257, 1285, 4369, 21845], the divisors 1, 5, 85 and 21845 are in A001045, thus a(21845) = 1 + 5 + 85 + 21845 = 21936.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to sums of divisors

Cf. A000203, A001045, A147612, A293431, A293434.

Cf. also A005092.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, # &, 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));
A293432(n) = sumdiv(n,d,A147612(d)*d);

from sympy import divisors
def A293432(n): return sum(d 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)*d.

a(n) = A293434(n) + (A147612(n)*n).

A293436 a(n) is the sum of the proper divisors of n that are Fibonacci numbers (A000045).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 3, 4, 8, 1, 6, 1, 3, 9, 11, 1, 6, 1, 8, 4, 3, 1, 14, 6, 16, 4, 3, 1, 11, 1, 11, 4, 3, 6, 6, 1, 3, 17, 16, 1, 27, 1, 3, 9, 3, 1, 14, 1, 8, 4, 16, 1, 6, 6, 11, 4, 3, 1, 11, 1, 3, 25, 11, 19, 6, 1, 37, 4, 8, 1, 14, 1, 3, 9, 3, 1, 19, 1, 16, 4, 3, 1, 27, 6, 3, 4, 11, 1, 11, 14, 3, 4, 3, 6, 14, 1, 3, 4, 8, 1, 40, 1, 24, 30

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 55, its proper divisors are [1, 5, 11], of which only 1 and 5 are in A000045, thus a(55) = 1 + 5 = 6.
For n = 10946, its proper divisors are [1, 2, 13, 26, 421, 842, 5473], and only 1, 2 and 13 are Fibonacci numbers, thus a(10946) = 1 + 2 + 13 = 16.

Antti Karttunen, Table of n, a(n) for n = 1..10946
Index entries for sequences related to sums of divisors

Cf. A000045, A005092, A010056, A293435.

Cf. also A293434.

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, # &, 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
A293436(n) = sumdiv(n,d,(dA010056(d)*d);

a(n) = Sum_{d|n, dA010056(d)*d.
a(n) = A005092(n) - (A010056(n)*n).
G.f.: Sum_{k>=2} Fibonacci(k) * x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021

A005093 Sum of squares of Fibonacci numbers 1,2,3,5,... that divide n.

Original entry on oeis.org

1, 5, 10, 5, 26, 14, 1, 69, 10, 30, 1, 14, 170, 5, 35, 69, 1, 14, 1, 30, 451, 5, 1, 78, 26, 174, 10, 5, 1, 39, 1, 69, 10, 1161, 26, 14, 1, 5, 179, 94, 1, 455, 1, 5, 35, 5, 1, 78, 1, 30, 10, 174, 1, 14, 3051, 69, 10, 5, 1, 39

Offset: 1

N. J. A. Sloane

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..100 from Indranil Ghosh)

Cf. A000045, A005092.

nmax = 100; With[{fibs = Fibonacci[Range[2, Floor[Log[nmax*Sqrt[5]] / Log[GoldenRatio]] + 1]]}, Table[Total[Select[fibs, Divisible[n, #1] & ]^2], {n, 1, nmax}]] (* Harvey P. Dale, Apr 25 2011, fixed by Vaclav Kotesovec, Apr 29 2019 *)

from sympy import divisors, fibonacci
l = [fibonacci(n) for n in range(1, 21)]
def a(n):
    return sum(i**2 for i in divisors(n) if i in l)
print([a(n) for n in range(1, 101)])  # Indranil Ghosh, Mar 22 2017

G.f.: Sum_{k>=2} Fibonacci(k)^2*x^Fibonacci(k)/(1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Mar 21 2017

Offset changed from 0 to 1 by Ilya Gutkovskiy, Mar 21 2017

b-file corrected by Vaclav Kotesovec, Apr 29 2019

A339621 Sum of Fibonacci divisors of n^2 + 1.

Original entry on oeis.org

1, 3, 6, 8, 1, 16, 1, 8, 19, 3, 1, 3, 6, 42, 1, 3, 1, 8, 19, 3, 1, 50, 6, 8, 1, 3, 1, 8, 6, 3, 1, 16, 6, 8, 103, 3, 1, 8, 6, 3, 1, 3, 6, 8, 14, 3, 1, 55, 6, 3, 1, 3, 6, 8, 1, 126, 1, 21, 6, 3, 14, 3, 6, 8, 1, 3, 1, 8, 6, 3, 391, 3, 6, 21, 1, 3, 1, 8, 6, 3, 1, 37

Offset: 0

Michel Lagneau, Dec 10 2020

nonn

A Fibonacci divisor of a number k is a Fibonacci number that divides k. (The divisor 1 is only counted once.)
For n < 2*10^5, the subsequence of primes begins by 3, 19, 37, 97, 103, 131, 139, 239, 241, 283, 359, 487, 631, ...
The Fibonacci numbers of the sequence are 1, 3, 8, 21, 55, 144, 377, ...
Conjecture: If the sum of the Fibonacci divisors of m^2 + 1 is a Fibonacci number, then this number belongs to the sequence A001906(n) = F(2n) where F(n) is the Fibonacci sequence.
The sequence giving the least k such that the sum of Fibonacci divisors of k^2 + 1 is equal to F(2*n) for n > 0 begins with: 0, 1, 3, 57, 47, 15007, 1679553, ...

			a(3) = 8 because the divisors of 3^2 + 1 = 10 are {1, 2, 5, 10}, and the sum of the Fibonacci divisors is 1 + 2 + 5 = 8.

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

Cf. A000045, A001906, A002522, A005092, A005574, A339461.

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

Array[DivisorSum[#^2 + 1, # &, AnyTrue[Sqrt[5 #^2 + 4 {-1, 1}], IntegerQ] &] &, 82, 0] (* Michael De Vlieger, Dec 10 2020 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
a(n) = sumdiv(n^2+1, d, if (isfib(d), d)); \\ Michel Marcus, Dec 10 2020

a(A005574(n)) = 1 for n > 2.
a(n) = 3 when n^2 + 1 = 2*p, p prime and non-Fibonacci number.
a(n) = A005092(A002522(n)). - Michel Marcus, Aug 10 2022

A300895 L.g.f.: log(Product_{k>=2} (1 + x^Fibonacci(k))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 1, 4, -3, 6, -2, 1, 5, 4, -4, 1, -6, 14, 1, 9, -11, 1, -2, 1, -8, 25, 1, 1, 2, 6, -12, 4, -3, 1, -7, 1, -11, 4, 35, 6, -6, 1, 1, 17, 0, 1, -23, 1, -3, 9, 1, 1, -14, 1, -4, 4, -16, 1, -2, 61, 5, 4, 1, 1, -11, 1, 1, 25, -11, 19, -2, 1, -37, 4, -4, 1, 2, 1, 1, 9, -3, 1, -15, 1, -16, 4, 1, 1, -27, 6

Offset: 1

Ilya Gutkovskiy, Mar 14 2018

sign

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 - 2*x^6/6 + x^7/7 + 5*x^8/8 + 4*x^9/9 - 4*x^10/10 + ...
exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + 2*x^10 + ... + A000119(n)*x^n + ...

Cf. A000045, A000119, A005092, A005478.

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^Fibonacci[k]), {k, 2, 14}]], {x, 0, nmax}],x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[Fibonacci[k] x^Fibonacci[k]/(1 + x^Fibonacci[k]), {k, 2, 14}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(5 #^2 + 4)^(1/2)] || IntegerQ[(5 #^2 - 4)^(1/2)] &], {n, 85}]

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

a(n) = n + 1 if n is an odd prime Fibonacci number (A005478 except a(1) = 2).

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

Michel Marcus, Jul 16 2022

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

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

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 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
row(n) = select(isfib, divisors(n));

cp's OEIS Frontend

A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).

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A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).

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A293432 Sum of Jacobsthal numbers that divide n.

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A293436 a(n) is the sum of the proper divisors of n that are Fibonacci numbers (A000045).

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A005093 Sum of squares of Fibonacci numbers 1,2,3,5,... that divide n.

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A339621 Sum of Fibonacci divisors of n^2 + 1.

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A300895 L.g.f.: log(Product_{k>=2} (1 + x^Fibonacci(k))) = Sum_{n>=1} a(n)*x^n/n.