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A263161 Positive values of n such that A000071(n+2) is divisible by A000217(n).

%I A263161 #38 Sep 08 2022 08:46:14
%S A263161 1,240,600,768,1008,1200,1320,1800,2160,2688,2736,3000,3360,3888,4800,
%T A263161 5280,5520,6120,6479,6480,6720,6840,7320,7680,8208,8640,9000,9600,
%U A263161 9720,10368,11160,11663,12240,12288,13200,13248,13440,13680,14400,15120,15360,15456,16560,18048
%N A263161 Positive values of n such that A000071(n+2) is divisible by A000217(n).
%C A263161 Interestingly, the minimum value of a(n) - a(n-1) is 1. Is there a maximum value of a(n) - a(n-1)?
%C A263161 From _Robert Israel_, Oct 19 2015: (Start)
%C A263161 n is in the sequence if either n is odd and A001175(n) and A001175((n+1)/2) both divide n+1, or n is even and A001175(n/2) and A001175(n+1) both divide n.
%C A263161 Most of the terms of the sequence appear to fall in these categories. The first two that do not are 15456 and 41640.
%C A263161 In particular, if n = 2^j * 3^k * 5^m with j >= 4, k >= 1 and m >= 1, and n+1 is prime, then n is in the sequence.  There are believed to be infinitely many numbers of this form.  The first few are 240, 1200, 2160, 4800, 6480, 7680, 8640, 9600, 14400, 15360. (End)
%H A263161 Charles R Greathouse IV, <a href="/A263161/b263161.txt">Table of n, a(n) for n = 1..10000</a>
%e A263161 For n = 1, A000071(1+2) = 1 is divisible by A000217(1) = 1.
%p A263161 fmod:= proc(a, b) local A, t;
%p A263161   uses LinearAlgebra[Modular];
%p A263161   if b < 4295022903 then t:= integer[8] else t:= integer fi;
%p A263161   A:= Mod(b, <<1, 1>|<1, 0>>, t);
%p A263161   MatrixPower(b, A, a)[1, 2];
%p A263161 end proc:
%p A263161 filter:= n -> (fmod(n+2, n*(n+1)/2) = 1):
%p A263161 filter(1):= true:
%p A263161 select(filter, [$1..10^5]); # _Robert Israel_, Oct 19 2015
%t A263161 fQ[n_] := Mod[Fibonacci[n + 2] - 1, n (n + 1)/2] == 0; Select[Range@20000, fQ] (* _Bruno Berselli_, Oct 19 2015 - after _Robert G. Wilson v_ in A263225 *)
%o A263161 (PARI) for(n=1, 20000, if((fibonacci(n+2)-1) % (n*(n+1)/2) == 0, print1(n", ")));
%o A263161 (PARI) is(n)=((Mod([1,1;1,0],n*(n+1)/2))^(n+2))[1,2]==1 \\ _Charles R Greathouse IV_, Oct 19 2015
%o A263161 (Magma) [n: n in [1..20000] | IsDivisibleBy(Fibonacci(n+2)-1, n*(n+1) div 2)]; // _Bruno Berselli_, Oct 19 2015
%Y A263161 Cf. A000045, A000071, A000217, A001175, A263225.
%K A263161 nonn
%O A263161 1,2
%A A263161 _Altug Alkan_, Oct 11 2015