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A247937 Least integer m > n such that m + n divides F(m) + F(n), where F(k) refers to the Fibonacci number A000045(k). 16
5, 22, 9, 8, 8, 18, 10, 16, 21, 14, 35, 24, 17, 34, 21, 32, 20, 30, 31, 28, 87, 26, 47, 36, 28, 46, 63, 32, 80, 42, 151, 40, 75, 38, 38, 60, 113, 39, 51, 56, 109, 49, 307, 52, 63, 50, 50, 72, 101, 70, 57, 68, 97, 66, 58, 64, 93, 62, 191, 84 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: Let A be any integer not congruent to 3 modulo 6. Define u(0) = 0, u(1) = 1, and u(n+1) = A*u(n) + u(n-1) for n > 0. Then, for any integer n > 0, there are infinitely many positive integers m such that m + n divides u(m) + u(n).

This implies that a(n) exists for any n > 0.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(2) = 22 since 22 + 2 = 24 divides F(22) + F(2) = 17711 + 1 = 17712 = 24*738.

MATHEMATICA

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

CROSSREFS

Cf. A000045, A247824, A247940, A248032.

Sequence in context: A220002 A156860 A225846 * A343840 A270406 A209049

Adjacent sequences:  A247934 A247935 A247936 * A247938 A247939 A247940

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 27 2014

STATUS

approved

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Last modified July 30 17:21 EDT 2021. Contains 346359 sequences. (Running on oeis4.)