A330711 -id:A330711 - cp's OEIS frontend

A330713 Numbers k such that both k and k+1 are Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

1, 7475, 10205, 13740, 40754, 52479, 93044, 95984, 141911, 151487, 196416, 198255, 202824, 202895, 213920, 231552, 335535, 339744, 363320, 366876, 404719, 408680, 434259, 446480, 487710, 495159, 504440, 528408, 585599, 607410, 645560, 646575, 665567, 735020, 736280

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

Can 3 consecutive numbers be both Zeckendorf-Niven numbers and lazy-Fibonacci-Niven numbers? Equivalently, are there numbers that are both in A328210 and A328214?

			7475 is a term since A007895(7475) = 5 and A112310(7475) = 13 and both 5 and 13 are divisors of 7475, and A007895(7476) = 6 and A112310(7476) = 12 and both 6 and 12 are divisors of 7476.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A328209 and A328213.

Cf. A328208, A328210, A328212, A328214, A330711.

A331091 Positive negaFibonacci-Niven numbers k (A331085) such that -k is a negative negaFibonacci-Niven number (A331088).

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 27, 30, 36, 48, 55, 60, 72, 84, 90, 96, 100, 108, 110, 112, 116, 120, 144, 150, 156, 172, 176, 180, 184, 192, 196, 208, 228, 234, 240, 246, 252, 260, 264, 288, 300, 305, 320, 328, 330, 336, 340, 360, 372, 378, 384, 396, 400, 415, 420, 460, 468, 475, 480, 492

Offset: 1

Amiram Eldar, Jan 08 2020

Positive numbers k that are divisible by the number of terms in the negaFibonacci representations of both k and -k (A215022 and A215023, respectively).

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

Intersection of A331085 and A331088.

Cf. A215022, A215023, A330711.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]];
f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i];
negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s];
Select[Range[500], Divisible[#, negaFibTermsNum[#]] && Divisible[#, negaFibTermsNum[-#]] &]

A331827 Positive negabinary-Niven numbers k (A331728) such that -k is a negative negabinary-Niven number (A331819).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 28, 30, 32, 33, 36, 40, 42, 48, 54, 56, 60, 63, 64, 66, 68, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 120, 124, 126, 128, 129, 132, 136, 138, 140, 144, 150, 156, 160, 162, 168, 174, 175, 180, 186, 192, 198, 200

Offset: 1

Amiram Eldar, Jan 28 2020

Positive numbers k that are divisible by the sums of digits in the negabinary representations of both k and -k.

All the powers of 2 above 1 are terms.

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

Intersection of A331728 and A331819.

Cf. A330711, A331091.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; seqQ[n_] := And @@ (Divisible[n, negaBinWt[#]] & /@ {-n, n}); Select[Range[200], seqQ]

A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

Original entry on oeis.org

3, 4, 5, 7, 15, 22, 25, 26, 27, 35, 41, 47, 49, 50, 73, 74, 75, 87, 89, 95, 97, 98, 101, 107, 121, 122, 135, 145, 146, 147, 167, 193, 194, 195, 207, 215, 217, 218, 221, 227, 241, 242, 255, 275, 289, 290, 315, 327, 335, 337, 338, 347, 361, 362, 385, 386, 387, 395

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

Numbers of the form F(k) - 1 have the same Zeckendorf (A014417) and dual Zeckendorf (A104326) representations: alternating digits of 1 and 0 whose sum is floor((k - 1)/2). Thus, if k is in this sequence then F(k) - 1 is both a Zeckendorf-Niven number (A328208) and a lazy-Fibonacci-Niven number (A328212), i.e., A000071(a(n)) is in A330711.

			7 is in this sequence since F(7) - 1 = 13 - 1 = 12 is divisible by floor((7 - 1)/2) = 3. The Zeckendorf and dual Zeckendorf representations of 7 are both 1010, whose sum of digits, 2, divides 12. Thus 12 is both a Zeckendorf-Niven number and a lazy-Fibonacci-Niven number.

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

Cf. A000045, A000071, A328208, A328212, A330711.

Select[Range[3, 400], Divisible[Fibonacci[#] - 1, Floor[(# - 1)/2]] &]

A333621 Numbers that are divisible by the total number of 1's in both the Zeckendorf and the dual Zeckendorf representations of all their divisors (A300837 and A333618).

Original entry on oeis.org

1, 2, 4, 126, 416, 442, 3025, 4588, 9243, 10428, 11900, 15070, 18176, 19436, 20532, 26956, 28582, 32108, 33028, 35278, 35929, 37634, 47678, 50386, 61952, 69254, 74578, 88984, 93534, 95120, 96334, 100326, 102297, 142894, 144039, 145768, 147664, 152817, 163125, 183002

Offset: 1

Amiram Eldar, Mar 29 2020

			126 is a term since A300837(126) = 21 and A333618(126) = 7 are both divisors of 126.

Intersection of A333619 and A333620.

Cf. A007895, A112310, A300837, A330711, A333617, A333618.

zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];
fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeckSum[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
dualZeckDivDigSum[n_] := DivisorSum[n, dualZeckSum[#] &];
Select[Range[10^4], Divisible[#, zeckDivDigSum[#]] && Divisible[#, dualZeckDivDigSum[#]] &]

cp's OEIS Frontend

A330713 Numbers k such that both k and k+1 are Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

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A331091 Positive negaFibonacci-Niven numbers k (A331085) such that -k is a negative negaFibonacci-Niven number (A331088).

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A331827 Positive negabinary-Niven numbers k (A331728) such that -k is a negative negabinary-Niven number (A331819).

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A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

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A333621 Numbers that are divisible by the total number of 1's in both the Zeckendorf and the dual Zeckendorf representations of all their divisors (A300837 and A333618).

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