A000350 -id:A000350 - cp's OEIS frontend

A023172 Self-Fibonacci numbers: numbers k that divide Fibonacci(k).

1, 5, 12, 24, 25, 36, 48, 60, 72, 96, 108, 120, 125, 144, 168, 180, 192, 216, 240, 288, 300, 324, 336, 360, 384, 432, 480, 504, 540, 552, 576, 600, 612, 625, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1080, 1104, 1152, 1176, 1200, 1224, 1296, 1320

Offset: 1

David W. Wilson

nonn

Sequence contains all powers of 5, infinitely many multiples of 12 and other numbers (including some factors of Fibonacci(5^j), e.g., 75025).
If m is in this sequence then 5*m is (since 5*m divides 5*F(m) which in turn divides F(5*m)). Also, if m is in this sequence then F(m) is in this sequence (since if gcd(F(m),m)=m then gcd(F(F(m)),F(m)) = F(gcd(F(m),m)) = F(m)). - Max Alekseyev, Sep 20 2009
From Max Alekseyev, Nov 29 2010: (Start)
Every term greater than 1 is a multiple of 5 or 12.
Proof. Let n>1 divide Fibonacci number F(n) and let p be the smallest prime divisor of n.
If p=2, then 3|n implying further that 4|n. Hence, 12|n.
If p=5, then 5|n.
If p is different from 2 and 5, then p divides either F(p+1) or F(p-1) and thus p divides either F(gcd(n,p+1)) or F(gcd(n,p-1)). Minimality of p implies that gcd(n,p-1)=1 and gcd(n,p+1)=1 (notice that p+1 being prime implies p=2 which is not the case). Therefore, p divides F(1)=1, a contradiction to the existence of such p. (End)
Luca & Tron give an upper bound, see links. - Charles R Greathouse IV, Aug 04 2021

S. Wolfram, "A new kind of science", p. 891

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 500 terms from T. D. Noe, next 4600 terms from Lars Blomberg)
Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025. See p. 1.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
Tamas Lengyel, Divisibility Properties by Multisection, Dec 2000. Fib Q. 41 (1) 72
Florian Luca and Emanuele Tron, The Distribution of Self-Fibonacci Divisors, Proceedings of the Thirteenth Conference of the Canadian Number Theory Association (CNTA XIII), Ayşe Alaca, Şaban Alaca, and Kenneth Williams, ed. (2015), pp. 149-158. arXiv:1410.2489 [math.NT], 2014.
Chris Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000350. See A127787 for an essentially identical sequence.
Cf. A000045, A069104, A123976, A159051, A263112.
Cf. A128974 (12n does not divide Fibonacci(12n)). - Zak Seidov, Jan 10 2016

import Data.List (elemIndices)
a023172 n = a023172_list !! (n-1)
a023172_list =
   map (+ 1) $ elemIndices 0 $ zipWith mod (tail a000045_list) [1..]
-- Reinhard Zumkeller, Oct 13 2011

[n: n in [1..2*10^3] | Fibonacci(n) mod n eq 0 ]; // Vincenzo Librandi, Sep 17 2015

fmod:= proc(n,m) local M,t; uses LinearAlgebra:-Modular;
    if m <= 1 then return 0 fi;
    if m < 2^25 then t:= float[8] else t:= integer fi;
    M:= Mod(m,<<1,1>|<1,0>>,t);
    round(MatrixPower(m,M,n)[1,2])
end proc:
select(n -> fmod(n,n)=0, [$1..2000]); # Robert Israel, May 10 2016

a=0; b=1; c=1; Do[a=b; b=c; c=a+b; If[Mod[c, n]==0, Print[n]], {n, 3, 1500}]
Select[Range[1350], Mod[Fibonacci[ # ], # ]==0&] (* Harvey P. Dale *)

is(n)=((Mod([1,1;1,0],n))^n)[1,2]==0 \\ Charles R Greathouse IV, Feb 03 2014

Edited by Don Reble, Sep 07 2003

A038546 Numbers n such that n-th Fibonacci number has initial digits n.

Original entry on oeis.org

0, 1, 5, 43, 48, 53, 3301, 48515, 348422, 406665, 1200207, 6698641, 190821326, 2292141445, 257125021372, 5843866639660, 45173327533483, 46312809996150, 59358981837795, 129408997210988, 1450344802530203, 5710154240910003

Offset: 1

Jeff Burch

The Mathematica coding used by Robert G. Wilson v implements Binet's Fibonacci number formula as suggested by David W. Wilson and incorporates Benoit Cloitre's use of logarithms to achieve a further increase in speed.

Fixed points of A020344. - Alois P. Heinz, Jul 08 2022

			a(3)=43 since 43rd Fibonacci number starts with 43 -> {43}3494437.
Fibonacci(53) is 53316291173, which begins with 53, so 53 is a term in the sequence.

Ron Knott, Fibonacci Numbers and the Golden Section
Eric Weisstein's World of Mathematics, Fibonacci numbers

Cf. A000045, A020344, A052000, A000350, A050816.

a = N[ Log[10, Sqrt[5]/5], 24]; b = N [Log[10, GoldenRatio], 24]; Do[ If[ IntegerPart[10^FractionalPart[a + n*b]*10^Floor[ Log[10, n]]] == n, Print[n]], {n, 225000000}] (* Robert G. Wilson v, May 09 2005 *)
(* confirmed with: *) fQ[n_] := (FromDigits[ Take[ IntegerDigits[ Fibonacci[n]], Floor[ Log[10, n] + 1]]] == n)

/* To obtain terms > 5: */ a=(1+sqrt(5))/2; b=1/sqrt(5); for(n=1,3500, if(n==floor(b*(a^n)/10^( floor(log(b *(a^n))/log(10))-floor(log(n)/log(10)))),print1(n,","))) \\ Benoit Cloitre, Feb 27 2002

n>5 is in the sequence if a=(1+sqrt(5))/2 b=1/sqrt(5) and n==floor(b*(a^n)/10^(floor((log(b) +n*log(a))/log(10))-floor(log(n)/log(10))) ). - Benoit Cloitre, Feb 27 2002

Term a(6) from Patrick De Geest, Oct 15 1999
a(7) from Benoit Cloitre, Feb 27 2002
a(8)-a(11) from Robert G. Wilson v, May 09 2005
a(12) from Robert G. Wilson v, May 11 2005
More terms from Robert Gerbicz, Aug 22 2006

A050816 Fibonacci(k) ending with digits of its index number k.

Original entry on oeis.org

0, 1, 5, 75025, 514229, 165580141, 7778742049, 2504730781961, 17167680177565, 259695496911122585, 1779979416004714189, 573147844013817084101, 59425114757512643212875125, 898923707008479989274290850145

Offset: 0

Patrick De Geest, Oct 15 1999

Cf. A000045, A000350, A038546, A052000.

fQ[n_]:=Module[{idni=IntegerDigits[First[n]],indf=IntegerDigits[Last[n]]}, idni== Take[indf,-Length[idni]]]; Transpose[Select[Table[ {n, Fibonacci[n]}, {n,0,150}],fQ]][[2]] (* Harvey P. Dale, Apr 21 2011 *)

A052000 Fibonacci(k) starting with digits of its index number k.

Original entry on oeis.org

0, 1, 5, 433494437, 4807526976, 53316291173

Offset: 1

Patrick De Geest, Oct 15 1999

Next term '3301...7801' has 690 digits.

Cf. A000045, A038546, A000350, A050816.

a(n) = A000045(A038546(n))

Offset changed to 1 by Jon E. Schoenfield, Oct 17 2019

A093537 Number of n-digit Fibomorphic numbers, i.e., numbers m such that Fibonacci(m) ends in m.

Original entry on oeis.org

3, 8, 22, 82, 228, 229, 231, 231, 230, 231, 232, 231, 230

Offset: 1

Lekraj Beedassy, May 14 2004

The sequence of k such that Fibonacci(k) mod 10^n = k mod 10^n has for n=1..10 the periods: 14, 31, 71, 271, 771, 771, 771, 771, 771, 771. This may help explain why the A093537 terms are almost constant for n>=5. [Lars Blomberg, Oct 02 2011]

			For n=1, there are 3 such 1-digit Fibonacci numbers: 0, 1 and 5.

Cf. A000350.

4 more terms from David Wasserman, Oct 26 2006

Offset changed to 1 and a(8)-a(13) added by Lars Blomberg, Oct 02 2011

A132365 Least number k such that the Lucas number A000032(k) contains n.

Original entry on oeis.org

1, 0, 2, 3, 13, 9, 4, 6, 7, 24, 5, 10, 15, 26, 20, 25, 49, 6, 11, 16, 13, 12, 10, 21, 45, 40, 20, 36, 7, 31, 50, 12, 35, 19, 17, 15, 41, 36, 22, 23, 39, 39, 14, 21, 41, 60, 8, 32, 19, 56, 20, 13, 45, 37, 51, 44, 17, 56, 42, 22, 25, 62, 35, 15, 71, 47, 25, 24, 43, 32, 17, 45, 49, 38

Offset: 1

Jonathan Vos Post, Nov 08 2007

Values such that a(n)=n (fixed points) are 1, 62. I don't know if there are any other fixed points. The first time a(n)=a(n+1) occurs because L(39)=141422324 which includes both 41 and 42 (and later on in the sequence, because it contains 141 and 142). [Sean A. Irvine, Nov 30 2009]

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000032, A000045, A000350, A050816, A038546, A052000, A023172.

def A132365(n):
    a, b, m, s = 2, 1, 0, str(n)
    while True:
        if s in str(a):
            return m
        m += 1
        a, b = b, a+b # Chai Wah Wu, Jun 06 2017

a(n) = Min{k such that A000032(k) contains the decimal digit substring which represents the integer n}.

Incorrect comment removed by Sean A. Irvine, Nov 30 2009

Corrected and extended by Sean A. Irvine, Nov 30 2009

cp's OEIS Frontend

A023172 Self-Fibonacci numbers: numbers k that divide Fibonacci(k).

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A038546 Numbers n such that n-th Fibonacci number has initial digits n.

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A050816 Fibonacci(k) ending with digits of its index number k.

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A052000 Fibonacci(k) starting with digits of its index number k.

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A093537 Number of n-digit Fibomorphic numbers, i.e., numbers m such that Fibonacci(m) ends in m.

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A132365 Least number k such that the Lucas number A000032(k) contains n.

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