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A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

%I A160665 #23 Jul 05 2025 09:55:02
%S A160665 1,3,24,48,72,96,120,144,192,216,240,288,336,360,384,406,432,480,576,
%T A160665 600,648,672,720,768,864,936,960,1008,1080,1104,1152,1200,1224,1296,
%U A160665 1320,1344,1368,1440,1536,1680,1728,1800,1920,1944,2016,2160,2208,2304
%N A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.
%C A160665 Numbers k such that Sum_{i=0..k} A000032(i)/(k+1) is an integer. - _Robert G. Wilson v_, May 25 2009
%C A160665 Why do the terms in A141767 so closely correspond to A160665? Except for k = 1, 3, 406, 44758, 341446, 1413286, 3170242, 4861698, 7912534, ..., k == 0 (mod 24). - _Robert G. Wilson v_, May 25 2009
%H A160665 Amiram Eldar, <a href="/A160665/b160665.txt">Table of n, a(n) for n = 1..10000</a>
%F A160665 {k: k | A001610(k)}. - _R. J. Mathar_, May 25 2009
%p A160665 A000032 := proc(n) option remember ; if n <= 1 then 2-n; else procname(n-1)+procname(n-2) ; fi; end: A001610 := proc(n) add(A000032(i),i=0..n-1) ; end: for n from 1 to 3000 do if A001610(n) mod n = 0 then printf("%d,",n) ; fi; od: # _R. J. Mathar_, May 25 2009
%t A160665 lst = {}; a = 2; b = 1; s = 3; n = 3; While[n < 2447, c = a + b; s = s + c; If[Mod[c, n] == 0, AppendTo[lst, n]]; a = b; b = c; n++ ]; lst (* _Robert G. Wilson v_, May 25 2009 *)
%Y A160665 Cf. A000032, A001610, A111058, A140826, A140971, A140972.
%K A160665 nonn
%O A160665 1,2
%A A160665 _Ctibor O. Zizka_, May 22 2009
%E A160665 More terms from _R. J. Mathar_ and _Robert G. Wilson v_, May 25 2009