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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

%I

%S 5,22,9,8,8,18,10,16,21,14,35,24,17,34,21,32,20,30,31,28,87,26,47,36,

%T 28,46,63,32,80,42,151,40,75,38,38,60,113,39,51,56,109,49,307,52,63,

%U 50,50,72,101,70,57,68,97,66,58,64,93,62,191,84

%N Least integer m > n such that m + n divides F(m) + F(n), where F(k) refers to the Fibonacci number A000045(k).

%C 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).

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

%H Zhi-Wei Sun, <a href="/A247937/b247937.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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}]

%Y Cf. A000045, A247824, A247940, A248032.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Sep 27 2014

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Last modified September 27 18:31 EDT 2021. Contains 347694 sequences. (Running on oeis4.)