A331085 -id:A331085 - cp's OEIS frontend

A331086 Positive numbers k such that k and k + 1 are both negaFibonacci-Niven numbers (A331085).

1, 4, 5, 9, 12, 13, 26, 68, 86, 87, 88, 89, 93, 99, 155, 176, 177, 183, 195, 212, 230, 231, 232, 233, 237, 243, 255, 320, 321, 327, 384, 395, 411, 415, 424, 464, 465, 471, 475, 484, 515, 544, 575, 591, 602, 644, 655, 656, 744, 824, 875, 894, 924, 1043, 1115, 1127

Offset: 1

Amiram Eldar, Jan 08 2020

Fibonacci numbers F(6*k - 1) and F(6*k) are terms.

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

Cf. A328209, A328213, A330927, A330931, A331085.

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];
negFibQ[n_] := Divisible[n, negaFibTermsNum[n]];
nConsec = 2; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec + 1; While[c < 55, If[And @@ neg, c++; AppendTo[seq, k - nConsec]];neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

A331087 Starts of runs of 3 consecutive positive negaFibonacci-Niven numbers (A331085).

Original entry on oeis.org

4, 12, 86, 87, 88, 176, 230, 231, 232, 320, 464, 655, 1194, 1592, 1596, 1854, 1914, 2815, 3016, 3294, 4124, 4178, 4179, 4180, 4268, 4412, 5663, 5755, 8360, 9894, 10614, 10703, 10915, 10975, 13936, 14994, 15114, 15714, 17630, 18976, 19984, 20824, 21835, 23175, 23513

Offset: 1

Amiram Eldar, Jan 08 2020

Numbers of the form F(6*k + 1) - 1, where F(m) is the m-th Fibonacci number, are terms.

Numbers of the form F(k) - 3, where k is congruent to {5, 11, 13, 19} mod 24 (A269819) are starts of runs of 5 consecutive negaFibonacci-Niven numbers.

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

Cf. A000045, A154701, A269819, A328210, A328214, A330932, A331085.

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];
negFibQ[n_] := Divisible[n, negaFibTermsNum[n]];
nConsec = 3; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec + 1; While[c < 55, If[And @@ neg, c++; AppendTo[seq, k - nConsec]];neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

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[-#]] &]

A331092 Positive numbers k such that k and k + 1 are both positive negaFibonacci-Niven numbers (A331085) and -k and -(k + 1) are both negative negaFibonacci-Niven numbers (A331088).

Original entry on oeis.org

1, 744, 875, 924, 1115, 1575, 1704, 1955, 2904, 3815, 5495, 5844, 6125, 6335, 6824, 7136, 7314, 8154, 8225, 8360, 8784, 9414, 10535, 10744, 10935, 11976, 12047, 13194, 13404, 13475, 18024, 19368, 19943, 20615, 21791, 22224, 22560, 23807, 24143, 24576, 25752, 26424, 26999

Offset: 1

Amiram Eldar, Jan 08 2020

Positive numbers k such that both k and k + 1 are in A331091.

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

Intersection of A331086 and A331089.

Cf. A331085, A331088, A331091.

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];
negFibQ[n_] := Divisible[n, negaFibTermsNum[n]] && Divisible[n, negaFibTermsNum[-n]];
nConsec = 2; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec + 1; While[c < 45, If[And @@ neg, c++; AppendTo[seq, k - nConsec]]; neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

A331728 Negabinary-Niven numbers: numbers divisible by the sum of digits in their negabinary representation (A027615).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42, 48, 50, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 69, 72, 76, 78, 80, 81, 84, 88, 90, 91, 95, 96, 100, 102, 108, 110, 112, 114, 120, 124, 125, 126, 128, 129, 132, 136, 138, 140

Offset: 1

Amiram Eldar, Jan 27 2020

			6 is a term since A039724(6) = 11010 and 1 + 1 + 0 + 1 + 0 = 3 is a divisor of 6.

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

Cf. A027615, A039724, A005349, A049445, A328208, A328212, A331085, A331088.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[n]]; Select[Range[100], negaBinNivenQ]

A342426 Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 (A244040).

Original entry on oeis.org

1, 2, 6, 9, 14, 21, 40, 42, 56, 72, 84, 108, 110, 120, 126, 130, 143, 154, 156, 162, 165, 168, 169, 176, 180, 182, 189, 198, 220, 225, 231, 243, 252, 280, 288, 297, 306, 308, 320, 322, 330, 336, 348, 350, 364, 390, 423, 430, 432, 459, 460, 462, 480, 490, 504

Offset: 1

Amiram Eldar, Mar 11 2021

			6 is a term since its representation in base 3/2 is 210 and 2 + 1 + 0 = 3 is a divisor of 6.
9 is a term since its representation in base 3/2 is 2100 and 2 + 1 + 0 + 0 = 3 is a divisor of 9.

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

Cf. A024629, A244040.
Subsequences: A342427, A342428, A342429.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary).

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[500], q]

A342726 Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 60, 64, 65, 66, 70, 77, 80, 88, 90, 96, 99, 100, 110, 112, 120, 124, 125, 126, 130, 140, 144, 145, 147, 150, 156, 160, 168, 170, 180, 182, 184, 185, 186, 190, 192

Offset: 1

Amiram Eldar, Mar 19 2021

Numbers k that are divisible by A066323(k).

Equivalently, Niven numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.

			2 is a term since its representation in base i-1 is 1100 and 1+1+0+0 = 2 is a divisor of 2.
10 is a term since its representation in base i-1 is 111001100 and 1+1+1+0+0+1+1+0+0 = 5 is a divisor of 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A007608, A066321, A066323, A271472, A342725, A342727, A342728, A342729.

Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary), A342426 (base 3/2).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := Divisible[n, Total[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]]; Select[Range[200], q]

A344341 Gray-code Niven numbers: numbers divisible by the number of 1's in their binary reflected Gray code (A005811).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 20, 24, 27, 28, 30, 31, 32, 33, 36, 39, 40, 42, 44, 45, 48, 51, 52, 56, 57, 60, 62, 63, 64, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 105, 108, 111, 112, 116, 120, 123, 124, 126, 127, 128, 129, 132, 135, 136

Offset: 1

Amiram Eldar, May 15 2021

			2 is a term since its Gray code is 11 and 1+1 = 2 is a divisor of 2.
6 is a term since its Gray code is 101 and 1+0+1 = 2 is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gray Code.
Wikipedia, Gray code.

Cf. A005811, A014550.
Subsequences: A344342, A344343, A344344.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary), A342426 (base 3/2), A342726 (base i-1).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[150], gcNivenQ]

A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 18, 20, 22, 24, 27, 29, 30, 32, 36, 39, 40, 42, 47, 48, 50, 54, 57, 58, 60, 64, 66, 69, 72, 76, 78, 80, 81, 84, 90, 92, 94, 96, 100, 104, 108, 120, 123, 124, 126, 129, 130, 132, 134, 135, 138, 140, 144, 152, 153, 156, 159, 160

Offset: 1

Amiram Eldar, Feb 17 2022

Numbers k such that A116543(k) | k.

			6 is a term since its minimal Lucas representation, A130310(6) = 1001, has A116543(6) = 2 1's and 6 is divisible by 2.

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

Cf. A000032, A116543, A130310.
Subsequences: A351715, A351716.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351719.

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[160], lucasNivenQ]

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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A331086 Positive numbers k such that k and k + 1 are both negaFibonacci-Niven numbers (A331085).

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A331087 Starts of runs of 3 consecutive positive negaFibonacci-Niven numbers (A331085).

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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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A331092 Positive numbers k such that k and k + 1 are both positive negaFibonacci-Niven numbers (A331085) and -k and -(k + 1) are both negative negaFibonacci-Niven numbers (A331088).

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A331728 Negabinary-Niven numbers: numbers divisible by the sum of digits in their negabinary representation (A027615).

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A342426 Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 (A244040).

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A342726 Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.

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A344341 Gray-code Niven numbers: numbers divisible by the number of 1's in their binary reflected Gray code (A005811).

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A351714 Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers (A130310).

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