A005086 -id:A005086 - cp's OEIS frontend

A132125 Number of distinct Fibonacci divisors of the factorial of n.

1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

R. J. Mathar, Oct 31 2007

nonn

			a(8)=7 because 8!=40320=2^7*3^2*5*7 has the seven divisors 1, 2, 3, 5, 8, 21 and 144 which are also Fibonacci numbers.

Cf. A005086, A000142, A000045.

A005086 := proc(n) local a,i,f; a := 0 ; for i from 2 do f := combinat[fibonacci](i) ; if f > n then RETURN(a) ; fi ; if n mod f = 0 then a := a+1 ; fi ; od: end: A000142 := proc(n) n! ; end: A := proc(n) A005086(A000142(n)) ; end: seq(A(n),n=1..80);

ndf[n_]:=Length[Intersection[fibs,Divisors[n!]]]; fibs=Fibonacci[ Range[600]];Array[ndf,75] (* Harvey P. Dale, Jun 24 2017 *)

a(n) = A005086(A000142(n)).

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));

A386625 Rectangular array R, read by descending antidiagonals: (row m) = increasing sequence of numbers m*F(n), n>=2, without duplicates, where F = A000045 (Fibonacci numbers); see Comments.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 10, 15, 12, 8, 16, 24, 20, 25, 13, 26, 39, 32, 40, 18, 21, 42, 63, 52, 65, 30, 7, 34, 68, 102, 84, 105, 48, 14, 64, 55, 110, 165, 136, 170, 78, 35, 104, 27, 89, 178, 267, 220, 275, 126, 56, 168, 45, 50, 144, 288, 432, 356, 445, 204, 91, 272, 72, 80, 11

Offset: 1

Clark Kimberling, Jul 27 2025

Every positive integer appears exactly once. Every non-Fibonacci prime appears in the first column. Except for initial terms, every row is a row of the Wythoff array, A035513.

			Corner:
   1    2    3    5    8   13   21   34   55
   4    6   10   16   26   42   68  110  178
   9   15   24   39   63  102  165  267  432
  12   20   32   52   84  136  220  356  576
  25   40   65  105  170  275  445  720 1165
  18   30   48   78  126  204  330  534  864
   7   14   35   56   91  147  238  385  623
  64  104  168  272  440  712 1152  864 3016
  27   45   72  117  189  306  495  801 1296
  50   80  130  210  340  550  890 1440 2330
  11   22   33   88  143  231  374  605  979
  36   60   96  156  252  408  660 1068 1728
Row 4 is obtained from 4*(row 1) by removing 4 and 8.

Cf. A000045, A005086 (counts removed duplicates), A035513, A386214.

ClearAll[rArray]
rArray[rows_, cols_] := Module[{fibs, R = {}, used = <||>, row, val, f, m},
   fibs = Map[Fibonacci, Range[2, cols + rows]];
   Table[row = {};
    Do[val = m*f; If[! KeyExistsQ[used, val], AppendTo[row, val];
      used[val] = True;], {f, fibs}]; Take[row, cols], {m, rows}]];
Grid[rArray[16, 12], Frame -> All]  (* array *)
r[m_, n_] := rArray[12, 12][[m]][[n]];
Table[r[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
(* Peter J. C. Moses, Jul 27 2025 *)

(row 1) = (1,2,3,5,8,13,...) = Fibonacci numbers, {F(n), n>=2}, and for m>1,

(row m) = numbers m*F(n) that are not h*F(k) for any h=2.

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A132125 Number of distinct Fibonacci divisors of the factorial of n.

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

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A386625 Rectangular array R, read by descending antidiagonals: (row m) = increasing sequence of numbers m*F(n), n>=2, without duplicates, where F = A000045 (Fibonacci numbers); see Comments.

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