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A297573 Least positive integer m such that m*n divides F(m+n), where F(k) denotes the k-th Fibonacci number A000045(k). 2
1, 1, 1, 2, 14170, 6, 1, 136, 207, 28340, 979, 12, 1, 322, 385, 368, 1, 306, 17, 19780, 3, 68, 1, 24, 524975, 58, 2889, 92, 13, 3570, 12749, 736, 7, 2, 165, 612, 1, 34, 633, 13160, 339, 6, 1, 1846, 5355, 2, 1, 336, 8183, 509950, 21, 116, 1, 918, 4895, 184, 51, 26, 10207, 7140 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

If p is a prime congruent to 2 or 3 modulo 5, then a(p) = 1 since it is known that p divides F(p+1).

Conjecture: a(n) exists for any n > 0.

See also A297574 for a similar conjecture.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..120 from Zhi-Wei Sun)

Zhi-Wei Sun, Introduction to Lucas sequences, a talk give in 2017.

EXAMPLE

a(2) = 1 since 1*2 divides F(1+2) = F(3) = 2.

a(4) = 2 since 2*4 divides F(2+4) = 8.

a(5) = 14170 since 5*14170 = 70850 divides F(5+14170) = F(14175).

a(6) = 6 since 6*6 = 36 divides F(6+6) = F(12) = 144.

MATHEMATICA

Do[m=1; Label[aa]; If[Mod[Fibonacci[m+n], m*n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

lpi[n_]:=Module[{k=1}, While[!Divisible[Fibonacci[k+n], k*n], k++]; k]; Array[ lpi, 60] (* Harvey P. Dale, May 05 2018 *)

PROG

(PARI) a(n) = my(m=1); while(1, if(Mod(fibonacci(m+n), m*n)==0, return(m)); m++) \\ Felix Fröhlich, Jan 01 2018

CROSSREFS

Cf. A000045, A247937, A248123, A297574.

Sequence in context: A070832 A170994 A151599 * A159730 A264944 A242855

Adjacent sequences:  A297570 A297571 A297572 * A297574 A297575 A297576

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 01 2018

STATUS

approved

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Last modified December 18 09:25 EST 2018. Contains 318219 sequences. (Running on oeis4.)