A130310 -id:A130310 - cp's OEIS frontend

A351715 Numbers k such that k and k + 1 are both Lucas-Niven numbers (A351714).

1, 2, 3, 6, 7, 10, 11, 29, 39, 47, 57, 80, 123, 129, 134, 152, 159, 170, 176, 199, 206, 245, 279, 326, 384, 387, 398, 404, 521, 531, 543, 560, 579, 615, 644, 651, 684, 755, 843, 849, 854, 872, 879, 890, 896, 944, 1024, 1052, 1064, 1070, 1071, 1095, 1350, 1382

Offset: 1

Amiram Eldar, Feb 17 2022

			6 is a term since 6 and 7 are both Lucas-Niven numbers: the minimal Lucas representation of 6, A130310(6) = 1001, has 2 1's and 6 is divisible by 2, and the minimal Lucas representation of 7, A130310(7) = 10000, has one 1 and 7 is divisible by 1.

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

Subsequence of A351714.
A351716 is a subsequence.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351720.

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]]]; Select[Range[1400], And @@ lucasNivenQ/@{#, #+1} &]

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

Original entry on oeis.org

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

Amiram Eldar, Feb 17 2022

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

			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.

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

Subsequence of A351714 and A351715.

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

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]

A351713 Numbers whose binary and minimal Lucas representations are both palindromic.

Original entry on oeis.org

0, 9, 31, 975, 297097, 816867, 4148165871, 152488124529, 1632977901693, 11162529166917, 11925833175477, 3047549778123957, 3894487365191355, 8920885515768255

Offset: 1

Amiram Eldar, Feb 17 2022

			   n    a(n)       A007088(a(n))                A130310(a(n))
   ----------------------------------------------------------
   1       0                   0                            0
   2       9                1001                        10001
   3      31               11111                     10000001
   4     975          1111001111              100010000010001
   5  297097 1001000100010001001  100001000000101000000100001

Index entries for sequences related to palindromes.

Intersection of A006995 and A351712.
Subsequence of A054770.
Cf. A007088, A130310.
Similar sequences: A095309, A331193, A331894, A351718.

lucasPalQ[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]]]; PalindromeQ[IntegerDigits[Total[2^s], 2]]]; Join[{0}, Select[Range[1, 10^6, 2], PalindromeQ[IntegerDigits[#, 2]] && lucasPalQ[#] &]]

A351718 Numbers whose binary and maximal Lucas representations are both palindromic.

Original entry on oeis.org

0, 3, 5, 17, 85, 107, 219, 1161, 1365, 1619, 2047, 4097, 6141, 19801, 25027, 68961, 91213, 134337, 1540157, 1804859, 11877549, 37696497, 44092437, 142710801, 548269377, 3387848595, 4073444175, 8226780335, 31029923047, 64662095631, 67947722943, 126590440407, 2145176968607

Offset: 1

Amiram Eldar, Feb 17 2022

			The first 10 terms are:
   n   a(n)  A007088(a(n))    A130311(a(n))
   ----------------------------------------
   1     0               0                0
   2     3              11               11
   3     5             101              101
   4    17           10001            11111
   5    85         1010101        101101101
   6   107         1101011        111010111
   7   219        11011011      10110101101
   8  1161     10010001001   11011111111011
   9  1365     10101010101  101010101010101
  10  1619     11001010011  101111010111101

Index entries for sequences related to palindromes.

Intersection of A006995 and A351717.

Cf. A007088, A130310, A351713.

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]]]; Join[{0}, Select[Position[s, _?PalindromeQ] // Flatten, PalindromeQ[IntegerDigits[#, 2]] &]]

A063732 Numbers whose Lucas representation excludes L_0 = 2.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90

Offset: 1

Fred Lunnon, Aug 25 2001

nonn

From Michel Dekking, Aug 26 2019: (Start)
This sequence is a generalized Beatty sequence.  We know that A054770, the sequence of numbers whose Lucas representation includes L_0=2, is equal to A054770(n) = A000201(n) + 2*n - 1 = floor((phi+2)*n) - 1.
One also easily checks that the numbers 3-phi and phi+2 form a Beatty pair. This implies that the sequence with terms floor((3-phi)*n)-1 is the complement of A054770 in the natural numbers 0,1,2,...
It follows that a(n) = 3*n - floor(n*phi) - 2.
(End)

Cf. A003622, A022342. Complement of A054770.
Partial sums of A003842.
Cf. A130310 (Lucas representation).

a(n) = floor((3-phi)*n)-1, where phi is the golden mean. - Michel Dekking, Aug 26 2019

A342089 Numbers that have two representations as the sum of distinct non-consecutive Lucas numbers (A000032).

Original entry on oeis.org

5, 12, 16, 23, 30, 34, 41, 45, 52, 59, 63, 70, 77, 81, 88, 92, 99, 106, 110, 117, 121, 128, 135, 139, 146, 153, 157, 164, 168, 175, 182, 186, 193, 200, 204, 211, 215, 222, 229, 233, 240, 244, 251, 258, 262, 269, 276, 280, 287, 291, 298, 305, 309, 316, 320, 327

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Brown (1969) proved that every positive number has a unique representation as a sum of non-consecutive Lucas numbers, if L(0) = 2 and L(2) = 3 do not appear simultaneously in the representation.
Chu et al. (2020) proved that if L(0) and L(2) are allowed to appear simultaneously, then each positive number can have at most two representations. The terms with two representations are listed in this sequence. They found that the number of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 17, 171, 1708, 17082, 170820, ..., and proved that the asymptotic density of this sequence is 1/(3*phi+1) = 0.1708203932... (A176015 - 1), where phi is the golden ratio (A001622).
A number n appears in the sequence if and only if the coefficient of phi^{-1} in the base-phi expansion of n is 1.  Alternatively, the last bit of the n-th term of A341722 is 1. - Jeffrey Shallit, May 03 2023

			5 is a term since it has two representations: L(0) + L(2) = 2 + 3 and L(1) + L(3) = 1 + 4.
12 is a term since it has two representations: L(1) + L(5) = 1 + 11 and L(0) + L(2) + L(4) = 2 + 3 + 7.

Robert Israel, Table of n, a(n) for n = 1..10000
John L. Brown, Jr., Unique representation of integers as sums of distinct Lucas numbers, The Fibonacci Quarterly, Vol. 7, No. 3 (1969), pp. 243-252.
Hung V. Chu, David C. Luo and Steven J. Miller, On Zeckendorf Related Partitions Using the Lucas Sequence, arXiv:2004.08316 [math.NT], 2020-2021.
David C. Luo, Java code for calculating non-consecutive partitions of natural numbers in any infinite integer sequence given by a second-order linear recurrence, GitHub.
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Cf. A000032, A001622, A130310, A116543, A176015, A341722.

Java
```
See David C. Luo's GitHub link.
```

L:= [seq(combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1), n=0..40)]:
f1:= proc(n, m) option remember;
      if n = 0 then return 1 fi;
      if m <= 0 then 0
      elif L[m] <= n then procname(n - L[m],m-2) + procname(n, m-1)
      else procname(n,m-1)
      fi
end proc:
filter:= n -> f1(n,ListTools:-BinaryPlace(L,n+1))=2:
select(filter, [$1..1000]); # Robert Israel, Mar 10 2021

L = Table[Fibonacci[n+1] + Fibonacci[n-1], {n, 0, 40}];
f1[n_, m_] := f1[n, m] = If[n == 0, Return[1], Which[m <= 0, 0, L[[m]] <= n, f1[n-L[[m]], m-2] + f1[n, m-1], True, f1[n, m-1]]];
filterQ[n_] := f1[n, FirstPosition[L, b_ /; b > n+1][[1]]-1] == 2;
Select[Range[1000], filterQ] (* Jean-François Alcover, Aug 27 2022, after Robert Israel *)

cp's OEIS Frontend

A351715 Numbers k such that k and k + 1 are both Lucas-Niven numbers (A351714).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A351713 Numbers whose binary and minimal Lucas representations are both palindromic.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A351718 Numbers whose binary and maximal Lucas representations are both palindromic.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A063732 Numbers whose Lucas representation excludes L_0 = 2.

Views

Author

Keywords

Comments

Crossrefs

Formula

A342089 Numbers that have two representations as the sum of distinct non-consecutive Lucas numbers (A000032).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

Maple

Mathematica