A131343 -id:A131343 - cp's OEIS frontend

A116543 Number of terms in greedy representation of n in terms of the Lucas numbers.

0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 2

Offset: 0

James E Davis, Mar 28 2006, Jun 07 2006

I have been studying A007895 and similar sequences and created this sequence as an analog of A007895 for the Lucas sequence (A000032).

			a(12)=2 because 12=11+1.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Cf. A000032, A007895, A007953, A130310, A131343.

s = Reverse[Sort[Table[LucasL[n - 1], {n, 1, 22}]]];
t = Map[Length[Select[Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, s]][[2,1]], # > 0 &]] &, Range[1000]] (* Peter J. C. Moses, Oct 18 2012 *)

Let L(n) = max(Lucas numbers < n). Then a(0) = 0, a(n) = 1 + a(n-L(n)).

a(n) = A007953(A130310(n)). - Amiram Eldar, Feb 17 2022

Edited by N. J. A. Sloane, Aug 10 2007

a(0) added by Amiram Eldar, Feb 17 2022

A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

1, 175, 216, 399, 656, 729, 737, 759, 1000, 1991, 2716, 2820, 2925, 3970, 4068, 4224, 4499, 4641, 5316, 5819, 6565, 6720, 6902, 7890, 9840, 10751, 11843, 12194, 12614, 13034, 13272, 14909, 15483, 15495, 16029, 17234, 17444, 17731, 18074, 18885, 19305, 19669, 20188

Offset: 1

Amiram Eldar, Feb 17 2022

			175 is a term since 175 and 176 are both lazy-Lucas-Niven numbers: the maximal Lucas representation of 175, A130311(175) = 1110110101, has 7 1's and 175 is divisible by 5, and the maximal Lucas representation of 176, A130311(7) = 1110110111, has 8 1's and 176 is divisible by 8.

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

Cf. A000032, A130311, A131343.
Subsequence of A351719.
A351721 is a subsequence.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715.

lazy = Select[IntegerDigits[Range[10^6], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[#*Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; SequencePosition[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], {True, True}][[;; , 1]]

A351721 Starts of runs of 3 consecutive lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

607068, 640618, 665720, 900921, 1000880, 1375940, 1505878, 1537250, 1924224, 1938508, 1966338, 2527998, 3394224, 3935424, 4242624, 4476624, 4747224, 4794624, 5351367, 5401824, 5526024, 5636356, 5992298, 6103900, 6343298, 7028362, 7113024, 8879424, 8998262, 9431424

Offset: 1

Amiram Eldar, Feb 17 2022

Conjecture: There are no runs of 4 consecutive lazy-Lucas-Niven numbers (checked up to 3*10^9).

			607068 is a term since 607068, 607069 and 607070 are all divisible by the number of terms in their maximal representation:
     k                   A130311(k)  A131343(k)  k/A131343(k)
-------------------------------------------------------------
607068  111010101010101011110111101         18          33726
607069  111010101010101011110111111         19          31951
607070  111010101010101011111010110         17          35710

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000032, A130311, A131343.
Subsequence of A351719 and A351720.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716.

A130311 Maximal (or "lazy") Lucas representation of n. Binary system for representing integers using Lucas numbers (A000032) as a base.

Original entry on oeis.org

0, 10, 1, 11, 110, 101, 111, 1011, 1110, 1101, 1111, 10110, 10101, 10111, 11011, 11110, 11101, 11111, 101011, 101110, 101101, 101111, 110110, 110101, 110111, 111011, 111110, 111101, 111111, 1010110, 1010101, 1010111, 1011011, 1011110, 1011101, 1011111, 1101011, 1101110

Offset: 0

Casey Mongoven, May 21 2007; corrected Mar 23 2008

			a(7) = 1110 because 4 + 3 + 1 = 8.
a(8) = 1101 because 4 + 3 + 2 = 9.

Edouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège, Vol. 41 (1972), pp. 179-182.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Ron Knott, Using Fibonacci Numbers to Represent Whole Numbers.

Cf. A000032, A104326, A130310, A131343.

lazy = Select[IntegerDigits[Range[10^2], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; Join[{0}, FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]] (* Amiram Eldar, Feb 17 2022 *)

a(0) and more terms from Amiram Eldar, Feb 17 2022

A351719 Lazy-Lucas-Niven numbers: numbers divisible by the number of terms in their maximal (or lazy) representation in terms of the Lucas numbers (A130311).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 40, 42, 54, 60, 66, 78, 84, 91, 96, 104, 112, 120, 126, 144, 154, 161, 168, 175, 176, 180, 182, 184, 192, 203, 210, 216, 217, 224, 232, 234, 240, 243, 264, 270, 280, 288, 304, 306, 310, 315, 320, 322, 328, 336, 344, 350, 360, 378

Offset: 1

Amiram Eldar, Feb 17 2022

Numbers k such that A131343(k) | k.

			6 is a term since its maximal Lucas representation, A130311(6) = 111, has A131343(6) = 3 1's and 6 is divisible by 3.

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

Cf. A000032, A130311, A131343.
Subsequences: A351720, A351721.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714.

lazy = Select[IntegerDigits[Range[3000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Position[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], True] // Flatten

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A116543 Number of terms in greedy representation of n in terms of the Lucas numbers.

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A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

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A351721 Starts of runs of 3 consecutive lazy-Lucas-Niven numbers (A351719).

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A130311 Maximal (or "lazy") Lucas representation of n. Binary system for representing integers using Lucas numbers (A000032) as a base.

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A351719 Lazy-Lucas-Niven numbers: numbers divisible by the number of terms in their maximal (or lazy) representation in terms of the Lucas numbers (A130311).

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