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%I #8 Feb 21 2022 02:30:06
%S 1,2,6,10,1070,4214,10654,10730,13118,31143,39830,43864,47663,48184,
%T 50134,62334,63510,79954,83344,84006,89614,107270,119224,119434,
%U 121384,124586,124984,129094,129843,148910,165430,167760,168574,183274,193144,198184,198904,199870
%N Starts of runs of 3 consecutive Lucas-Niven numbers (A351714).
%C Conjecture: 1 is the only start of a run of 4 consecutive Lucas-Niven numbers (checked up to 10^9).
%H Amiram Eldar, <a href="/A351716/b351716.txt">Table of n, a(n) for n = 1..10000</a>
%e 6 is a term since 6, 7 and 8 are all Lucas-Niven numbers: the minimal Lucas representation of 6, A130310(6) = 1001, has 2 1's and 6 is divisible by 2, the minimal Lucas representation of 7, A130310(7) = 10000, has one 1 and 7 is divisible by 1, and the minimal Lucas representation of 8, A130310(8) = 10010, has 2 1's and 8 is divisible by 2.
%t lucasNivenQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Divisible[n, Plus @@ IntegerDigits[Total[2^s], 2]]]; seq[count_, nConsec_] := Module[{luc = lucasNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ luc, c++; AppendTo[s, k - nConsec]]; luc = Join[Rest[luc], {lucasNivenQ[k]}]; k++]; s]; seq[50, 3]
%Y Subsequence of A351714 and A351715.
%Y Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351721.
%K nonn,base
%O 1,2
%A _Amiram Eldar_, Feb 17 2022
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