A072378 -id:A072378 - cp's OEIS frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

1, 6, 18, 54, 162, 486, 1458, 1926, 4374, 5778, 13122, 17334, 39366, 52002, 118098, 156006, 206082, 354294, 468018, 618246, 1062882, 1404054, 1854738, 2471058, 3188646, 4212162, 5564214, 7413174, 9565938, 12636486, 16692642, 22050774

Offset: 1

Robert G. Wilson v

nonn

Note that if n divides A000032(n) and p is an odd prime divisor of A000032(n), then pn divides A000032(pn) and, furthermore, p^k*n divides A000032(p^k*n) for every integer k>=0.
In particular, since 6 divides A000032(6) = 2*3^2, A016089 includes all terms of the geometric progression 2*3^k for k>0 (see A099856); since 18 divides A000032(18) = 2*3^3*107, A016089 includes all terms of the form 2*107^m*3^k for k>1 and m>=0; etc.
Terms of A016089 starting with 18 are multiples of 18. There are no other terms of the form 18p where p is prime, except for p=3 and p=107. - Alexander Adamchuk, May 11 2007

Lars Blomberg, Table of n, a(n) for n = 1..91
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A000204, A025192, A008776.

Cf. A099856, A072378 = numbers n such that 12n divides Fibonacci(12n), A023172 = numbers n such that n divides Fibonacci(n).

a = 1; b = 3; Do[c = a + b; a = b; b = c; If[Mod[c, n] == 0, Print[n]], {n, 3, 2, 10^6}]

is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Extended and revised by Max Alekseyev, May 13 2007, May 15 2008, May 16 2008

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

A124455 Numbers n such that 24n divides the sum of the first 24n nonzero Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 51, 54, 55, 56, 57, 60, 64, 70, 72, 75, 80, 81, 84, 86, 90, 92, 96, 98, 100, 102, 108, 110, 112, 114, 120, 125, 126, 128, 135, 138, 140, 144, 150, 153, 155, 160, 162

Offset: 1

Alexander Adamchuk, Nov 02 2006

nonn

Numbers n such that n divides the sum of the first n nonzero Fibonacci numbers are listed in A111035 = {1, 2, 24, 48, 72, 77, 96, 120, 144, 192, 216, 240, 288, 319, 323, 336, 360, ...}. Most of these are multiples of 24. Those which are not a multiple of 24 are listed in A124456 = {1, 2, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, ...}.
This sequence coincides with A072378 (12n | F(12n)) for all values up to 84. The first two different terms are 86 and 164.
Prime a(n) are {2, 3, 5, 281, ...}.

Cf. A111035, A124456, A072378 (numbers n such that 12n divides Fibonacci(12n)), A000045 (Fibonacci numbers).

Select[Range[10000], IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&] /24

Edited by M. F. Hasler, Feb 04 2020

A317971 Numbers m such that the Pisano period A001175(m) divides m.

Original entry on oeis.org

1, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2208, 2304, 2352, 2400, 2448, 2592, 2640, 2688, 2736, 2880, 3000, 3024, 3072

Offset: 1

N. J. A. Sloane, Sep 01 2018

nonn

More terms than usual are displayed because there are some very similar sequences in the OEIS.
The terms > 1 are divisible by 24, and the quotients give A072378.
In their paper, Shtefan and Dobrovolska (2018) prove that all terms m > 1 of this sequence are such that the sum of any m consecutive Fibonacci numbers is divisible by m. - Petros Hadjicostas, May 19 2019

N. J. A. Sloane, Table of n, a(n) for n = 1..10001
D. Shtefan and I. Dobrovolska, The sums of the consecutive Fibonacci numbers,, Fibonacci Quarterly, 56 (2018), 229-236.
Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001175, A072378.

A272355 Numbers of the form Fibonacci(12n)/(144n).

Original entry on oeis.org

1, 161, 34561, 8346401, 2150012161, 576914365601, 44861726436508961, 12840299190293644801, 3721082815965949056161, 321507757074243457409731361, 28572486227889263832443550935201, 8586901708088882505643582648796161

Offset: 1

Peter M. Chema, Apr 29 2016

The last two digits end either in 01 or 61. Digital root alternates 1 and 8.

Consecutive terms have ratios that approximate the product of Golden Ratio powers of multiples of 12 and consecutive integers fractions: E.g., the 4th term divided by the 3rd term approximates Golden Ratio^12 * 3/4; the 10th term divided by the 9th term approximates Golden Ratio^24 * 5/6; and the 16th term divided by the 15 term is a close approximation of Golden Ratio^48 * 5/6, etc.

			a(3) = Fib(12*3)/(144*3) = Fib36 / 432 = 34561; therefore, the third term is the integer 34561.

Cf. A072378.

Select[Table[Fibonacci[12n]/(144n),{n,20}],IntegerQ] (* Harvey P. Dale, Sep 26 2016 *)

for(n=1,100, t=fibonacci(12*n)/144/n; if(denominator(t)==1, print1(t", "))) \\ Charles R Greathouse IV, Apr 30 2016

a(n) = Integer Values of Fib(12n)/(144n)

cp's OEIS Frontend

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

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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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A124455 Numbers n such that 24n divides the sum of the first 24n nonzero Fibonacci numbers.

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A317971 Numbers m such that the Pisano period A001175(m) divides m.

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A272355 Numbers of the form Fibonacci(12n)/(144n).

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