A023172 -id:A023172 - cp's OEIS frontend

A128288 a(n) = A023163(n)/3 for n > 1.

3, 13, 37, 43, 53, 67, 83, 107, 157, 163, 173, 197, 227, 277, 283, 293, 307, 317, 347, 373, 397, 443, 467, 523, 547, 557, 563, 587, 613, 643, 653, 677, 683, 733, 757, 773, 787, 797, 827, 853, 877, 883, 907, 947, 997, 1013, 1093, 1117, 1123, 1163, 1187, 1213

Offset: 2

Alexander Adamchuk, Feb 24 2007

nonn

3 divides A023163(n) for n > 1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n). Almost all terms of {a(n)} are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} (primes congruent to {2, 3} mod 5) that are also the primes p that divide Fibonacci(p+1). The first composite term is a(74) = 1853 = 17*109. The second composite term is 9701 = 89*109. The third composite term is 10877 = 73*149 belong to A069107(n) Composite n such that n divides F(n+1) where F(k) are the Fibonacci numbers. Composite terms in {a(n)} are listed in A128289 = {1853, 9701, 10877, 17261, ...}.

			A023163 begins {1, 9, 39, 111, 129, 159, 201, 249, 321, 471, 489, 519, ...}.
Thus a(2) = A023163(2)/3 = 9/3 = 3, a(3) = A023163(3)/3 = 39/3 = 13.

Cf. A002708, A023172, A023173, A023162, A023163 (numbers k such that Fibonacci(k) == -2 (mod k)).

Cf. A003631, A069107, A128289 (composite terms in A128288).

a(n) = A023163(n)/3 for n > 1.

A128974 Numbers k such that 12k does not divide Fibonacci(12k).

Original entry on oeis.org

7, 11, 13, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 52, 53, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 99, 101, 103, 104, 105, 106, 107, 109, 111, 113, 115, 116, 117

Offset: 1

Alexander Adamchuk, May 11 2007

nonn

Complement of A072378 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 25, 27, ...} (numbers k such that 12k divides Fibonacci(12k)). It appears that {a(n)} includes all powers p^k for prime p > 5 and integer k > 0.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A072378 (numbers k such that 12k divides Fibonacci(12k)).

Cf. A023172 (numbers k such that k divides Fibonacci(k)).

Select[ Range[400], !IntegerQ[ Fibonacci[ 12# ] / (12#) ] & ]

A129066 Numbers k such that k divides Fibonacci(k) with multiples of 12 excluded.

Original entry on oeis.org

1, 5, 25, 125, 625, 3125, 15625, 75025, 78125, 375125, 390625, 1875625, 1953125, 9378125, 9765625, 46890625, 48828125, 234453125, 244140625, 332813125, 1172265625, 1220703125, 1664065625, 5628750625, 5861328125, 6103515625, 8320328125, 9006076025

Offset: 1

Alexander Adamchuk, May 11 2007

nonn

Set difference of A023172 and 12*A072378.
The sequence is closed under multiplication.
Also, if m is in this sequence (i.e., gcd(F(m),m)=m) then F(m) is in this sequence (since gcd(F(F(m)),F(m)) = F(gcd(F(m),m)) = F(m)).
In particular, this sequence includes all terms of geometric progressions 5^k*Fibonacci(5^m) for integers k >= 0 and m >= 0.

			a(1) = Fibonacci(1) = 1,
a(2) = Fibonacci(5) = 5,
a(3)..a(7) = {5^2, 5^3, 5^4, 5^5, 5^6},
a(8) = 75025 = 5^2*3001 = Fibonacci(5^2),
a(9) = 5^7,
a(10) = 375125 = 5^3*3001 = 5*Fibonacci(5^2),
a(11) = 5^8.

F. Lengyel, Divisibility Properties by Multisection, Fib. Quart. 41 (1) 72

Prime divisors are given in A171980. Their smallest multiples are given in A171981.

Cf. A072378, A023172.

Do[ If[ !IntegerQ[ n/12 ] && IntegerQ[ Fibonacci[n] / n ], Print[n] ], {n,1,5^8} ]

is(n)=n%12 && (Mod([0,1;1,1],n)^n*[0;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Edited and extended by Max Alekseyev, Sep 20 2009
a(1)=1 added by Zak Seidov, Nov 01 2009
Edited and extended by Max Alekseyev, Jan 20 2010

A270313 Denominator of Fibonacci(n)/n.

Original entry on oeis.org

1, 2, 3, 4, 1, 3, 7, 8, 9, 2, 11, 1, 13, 14, 3, 16, 17, 9, 19, 4, 21, 22, 23, 1, 1, 26, 27, 28, 29, 3, 31, 32, 33, 34, 7, 1, 37, 38, 39, 8, 41, 21, 43, 44, 9, 46, 47, 1, 49, 2, 51, 52, 53, 27, 11, 8, 57, 58, 59, 1, 61, 62, 63, 64, 13, 33, 67, 68, 69, 14, 71, 1, 73, 74, 3

Offset: 1

Jean-François Alcover and Paul Curtz, Mar 15 2016

a(n) = 1 for n in A023172; a(n) = n for n in A074215. - Robert Israel, Mar 16 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A023172, A074215, A104714, A127787, A270312 (numerators).

seq(n/igcd(n,combinat:-fibonacci(n)), n=1..100); # Robert Israel, Mar 16 2016

Table[Fibonacci[n]/n, {n, 1, 100}] // Denominator

a(n) = denominator(fibonacci(n)/n); \\ Michel Marcus, Mar 16 2016

a(n) = n/A104714(n). - Robert Israel, Mar 16 2016

A128289 Composite terms in A128288(n) = A023163(n)/3 for n>1.

Original entry on oeis.org

1853, 9701, 10877, 17261, 23323, 27403, 75077, 80189, 113573, 120581, 161027, 162133, 163059, 196877, 213749, 291941, 361397, 400987, 427549, 482677, 635627, 667589, 941291, 1030373, 1033997, 1140701, 1196061, 1256293, 1751747, 1816363, 1842581, 2288453, 2662277

Offset: 1

Alexander Adamchuk, Feb 24 2007

nonn

3 divides A023163(n) for n>1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n).
Almost all terms of A128288 are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} Primes congruent to {2, 3} mod 5; that are also the primes p that divide Fibonacci(p+1).
a(3) = 10877 = 73*149 belongs to A069107 Composite n such that n divides Fibonacci(n+1).
a(3) = 10877 and a(4) = 17261 belong to A094395 Odd composite n such that n divides Fibonacci(n) + 1.

			a(1) = A128288(74) = 1853 = 17*109.
a(2) = 9701 = 89*109.
a(3) = 10877 = 73*149.
a(4) = 17261 = 41*421.
a(5) = 23323 = 83*281.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A128288, A002708, A023172, A023173, A023162, A023163 = numbers n such that Fib(n) == -2 (mod n). Cf. A003631, A069107, A094413, A094395 = Odd composite n such that n divides Fibonacci(n) + 1.

Do[ f = Mod[ Fibonacci[3n], 3n ]; If[ !PrimeQ[n] && f == 3n-2, Print[ {n, FactorInteger[n]} ]], {n,1,25000} ]

Two more terms from R. J. Mathar, Oct 08 2007

a(9)-a(33) from Amiram Eldar, Apr 07 2019

A128935 a(n) = Fibonacci(5^n) / 5^n.

Original entry on oeis.org

1, 1, 3001, 475400918060101145703001, 29642179764875707696452732234250095350341524541114277856812964100763567848899514572925690068090872073476146381237687662210078001

Offset: 0

Alexander Adamchuk, May 11 2007

Numbers k such that k divides Fibonacci(k) are listed in A023172.
All powers of 5 belong to A023172.
5^n divides Fibonacci(5^n).
a(n) == 1 (mod 1000).
{a(n+1)/a(n)} = {1, 3001, 158414167964045700001, 62351961552434956321060201440347372028390478647963811251289490034177804212636326088548682319305439375001, ...}.

Cf. A023172, A121169, A121170, A058635, A045529.

a := proc(n) option remember; if n = 0 then 1 else 5^(4*n-3)*a(n-1)^5 - 5^(2*n-1)*a(n-1)^3 + a(n-1) end if; end proc: seq(a(n), n = 0..5); # Peter Bala, Nov 24 2022

Table[ Fibonacci[ 5^n ] / 5^n, {n,0,4} ]

a(n) = Fibonacci(5^n) / 5^n.

a(n+1) = 5^(4*n+1)*a(n)^5 - 5^(2*n+1)*a(n)^3 + a(n) with a(0) = 1. - Peter Bala, Nov 24 2022

A130163 Numbers k such that k^2 divides 2*Fibonacci(k).

Original entry on oeis.org

1, 12, 24, 168, 552, 2184, 3864, 4872, 13944, 28056, 35448, 47208, 50232, 63336, 70728, 75624, 76728, 112056, 172536, 181272, 224952, 239736, 254472, 287448, 320712, 364728, 381432, 404376, 457608, 460824, 508872, 529368, 537096, 613704, 645288, 813624

Offset: 1

Alexander Adamchuk, May 14 2007

nonn

A subset of A023172.
All listed terms for n>2 are divisible by a(3) = 24 = 2^3*3.
All listed terms for n>3, except a(5), are divisible by a(4) = 168 = 2^3*3*7.

			24 is a term because 24^2 = 2^6*3^2 divides 2*Fibonacci(24) = 2*46368 = 2^6*3^2*7*23.

Keith Schneider and Giovanni Resta, Table of n, a(n) for n = 1..776 (terms < 4*10^9, first 39 terms from Keith Schneider)

Cf. A000045.

Cf. A023172 (n divides Fibonacci(n)), A130164 (n^2 divides 3*Fibonacci(n)).

[n: n in [1..2*10^5] | 2*Fibonacci(n) mod n^2 eq 0 ]; // Vincenzo Librandi, Sep 17 2015

a=0; b=1; c=1; Do[a=b; b=c; c=a+b; If[Mod[2c,(n+2)^2]==0,Print[n+2]],{n,1,40000}] (* Stefan Steinerberger, May 15 2007 *)
A130163 = {1}; a = 0; b = 12; c = 3864; Do[If[Mod[24b, n^2] == 0, A130163 = Append[A130163, n]]; a = b; b = c; c = 322b - a;, {n, 12, 1000000, 12}];
A130163
Length[A130163]
(* Keith Schneider, May 27 2007 *)

More terms from Stefan Steinerberger, May 15 2007

a(14) corrected by N. J. A. Sloane, Nov 23 2007

A130164 Numbers k such that k^2 divides 3*Fibonacci(k).

Original entry on oeis.org

1, 12, 36, 612, 684, 3852, 11628, 25308, 41004, 65484, 73188, 77292, 155268, 156636, 250308, 430236, 467172, 545148, 562428, 779076, 977364, 1244196, 1313964, 1847484, 2123028, 2185452, 2621196, 2639556, 2662812, 2707956, 2859804, 3770892, 4387428, 4679244, 4755852, 4942116, 5744916, 5795532, 6394716, 7941924, 8053308, 8270244, 9267516

Offset: 1

Alexander Adamchuk, May 14 2007

nonn

A subset of A023172. All listed terms for n>1 are divisible by a(2) = 12 = 2^2*3. All listed terms for n>2 are divisible by a(3) = 36 = 2^2*3^2. - Robert G. Wilson v, May 15 2007

			36 is a term because 36^2 = 2^4*3^4 divides 3*Fibonacci(36) = 3*14930352 = 2^4*3^4*17*19*107.

Giovanni Resta, Table of n, a(n) for n = 1..289 (terms < 4*10^9)

Cf. A000045.

Cf. A023172 (n divides Fibonacci(n)), A130163 (n^2 divides 2*Fibonacci(n)).

[n: n in [1..2*10^5] | 3*Fibonacci(n) mod n^2 eq 0 ]; // Vincenzo Librandi, Sep 17 2015

a=0; b=1; c=1; Do[ a=b; b=c; c=a+b; If[ Mod[3c,(n+2)^2 ] == 0, Print[n+2]],{n, 1, 30000}] (* Stefan Steinerberger, May 15 2007 *)
a = 0; b = 0; c = 1; lst = {}; Do[ If[ Mod[3c, n^2] == 0, AppendTo[lst, n]]; a = b; b = c; c = a + b; {n, 2000000}]; lst (* Robert G. Wilson v *)
A130164 = {1}; a = 0; b = 12; c = 3864; Do[If[Mod[36b, n^2] == 0, A130164 = Append[A130164, n]]; a = b; b = c; c = 322b - a;, {n, 12, 1000000, 12}]; A130164
Length[A130164]
(* Keith Schneider, May 27 2007 *)

for(n=1,10^7,A=matrix(2,2,i,j,Mod(1,n*n)*(i+j<4))^n;if(lift(3*A[1,2])==0,print1(n",")))

More terms from Stefan Steinerberger and Robert G. Wilson v, May 15 2007

More terms from Robert Gerbicz, Nov 28 2010

A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).

Original entry on oeis.org

27, 807, 1707, 2977, 3027, 3277, 4717, 5137, 5677, 5917, 5967, 6187, 7087, 7357, 7597, 7707, 8217, 9117, 9297, 9387, 9667, 9877, 9927, 9997, 10387, 11097, 11647, 11797, 12727, 13407, 13867, 15757, 15987, 16327, 16507, 16857, 17347, 17767, 18237, 18817, 18997

Offset: 1

Arkadiusz Wesolowski, Apr 06 2009

nonn

Chris Caldwell, The Prime Glossary, Fibonacci number

Subsequence of A159259.

Cf. A000045, A023172, A061446, A159231, A181890.

lst = {1}; Do[f = Fibonacci[a]; Do[f = f/GCD[f, lst[[d]]], {d, Most[Divisors[a]]}]; AppendTo[lst, f], {a, 2, 19000}]; Flatten[Table[If[! PrimeQ[n] && Mod[lst[[n]], 8*n^2 - 2*n - 1] == 0, n, {}], {n, 19000}]] (* Arkadiusz Wesolowski, Dec 12 2011 *)

A167745 Integer values of Fibonacci(n)/n.

Original entry on oeis.org

1, 1, 12, 1932, 3001, 414732, 100156812, 25800145932, 6922972387212, 538340717238107532, 154083590283523737612, 44652993791591388673932, 475400918060101145703001, 3858093084890921488916776332

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 10 2009

nonn

A023172 [From Charles R Greathouse IV, Nov 12 2009]

f[n_]:=Fibonacci[n]/n; lst={};Do[If[IntegerQ[f[n]],AppendTo[lst,f[n]]],{n,6!}];lst
Select[Table[Fibonacci[n]/n,{n,200}],IntegerQ] (* Harvey P. Dale, Mar 02 2025 *)

cp's OEIS Frontend

A128288 a(n) = A023163(n)/3 for n > 1.

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A128974 Numbers k such that 12k does not divide Fibonacci(12k).

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A129066 Numbers k such that k divides Fibonacci(k) with multiples of 12 excluded.

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A270313 Denominator of Fibonacci(n)/n.

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A128289 Composite terms in A128288(n) = A023163(n)/3 for n>1.

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A128935 a(n) = Fibonacci(5^n) / 5^n.

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A130163 Numbers k such that k^2 divides 2*Fibonacci(k).

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Magma

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A130164 Numbers k such that k^2 divides 3*Fibonacci(k).

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A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).