A351716 Starts of runs of 3 consecutive Lucas-Niven numbers (A351714).

1, 2, 6, 10, 1070, 4214, 10654, 10730, 13118, 31143, 39830, 43864, 47663, 48184, 50134, 62334, 63510, 79954, 83344, 84006, 89614, 107270, 119224, 119434, 121384, 124586, 124984, 129094, 129843, 148910, 165430, 167760, 168574, 183274, 193144, 198184, 198904, 199870

Offset: 1

Author

Amiram Eldar, Feb 17 2022

Keywords

• nonn
• base

Comments

Conjecture: 1 is the only start of a run of 4 consecutive Lucas-Niven numbers (checked up to 10^9).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A351714 and A351715.

Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351721.

Programs

Mathematica

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]