A334308 -id:A334308 - cp's OEIS frontend

A334309 Numbers k such that k and k+1 are both base phi Niven numbers (A334308).

1, 15, 35, 90, 95, 231, 644, 728, 944, 1016, 1110, 1331, 1629, 1736, 1770, 1899, 1925, 2232, 2255, 2384, 2456, 2629, 2652, 2760, 3104, 3176, 3288, 3444, 3729, 3789, 3860, 4410, 4415, 4509, 4544, 4718, 4939, 4960, 5229, 5239, 5489, 5789, 5831, 5984, 6039, 6111

Offset: 1

Amiram Eldar, Apr 22 2020

			1 is a term since 1 and 2 are both base phi Niven numbers.

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

Cf. A328205, A328209, A328213, A330927, A330931, A333427, A334308.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; Select[Range[6000], phiNivenQ[#] && phiNivenQ[# + 1] &]

A334310 Starts of runs of 3 consecutive base phi Niven numbers (A334308).

Original entry on oeis.org

17171, 20760, 29183, 32772, 51336, 65840, 66608, 67990, 89054, 95563, 103682, 108910, 133990, 136512, 167598, 173640, 190094, 197218, 205478, 207364, 223873, 241934, 247115, 248443, 252014, 258816, 261135, 278783, 285129, 285130, 289392, 325934, 326520, 335178

Offset: 1

Amiram Eldar, Apr 22 2020

			17171 is a term since 17171, 17172 and 17173 are all base phi Niven numbers.

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

Cf. A154701, A328206, A328210, A328214, A330932, A333428, A334308, A334309.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; q1 = phiNivenQ[1]; q2 = phiNivenQ[2]; seq = {}; Do[q3 = phiNivenQ[n]; If[q1 && q2 && q3, AppendTo[seq, n - 2]]; q1 = q2; q2 = q3, {n, 3, 300000}]; seq

A334311 Starts of runs of 4 consecutive base phi Niven numbers (A334308).

Original entry on oeis.org

285129, 1958893, 2501533, 6488440, 7069840, 8803023, 16514327, 23826399, 34031773, 52256248, 68198847, 72969138, 76779087, 77622950, 87430210, 87474672, 96485487, 114137958, 120197293, 136275022, 151444458, 173740578, 174878352, 183872325, 188385855, 196268415

Offset: 1

Amiram Eldar, Apr 22 2020

			285129 is a term since 285129, 285130, 285131 and 285132 are all base phi Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..155

Cf. A141769, A330933, A331824, A331825, A334308, A334309, A334310.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; q1 = phiNivenQ[1]; q2 = phiNivenQ[2]; q3 = phiNivenQ[3]; seq = {}; Do[q4 = phiNivenQ[n]; If[q1 && q2 && q3 && q4, AppendTo[seq, n - 3]]; q1 = q2; q2 = q3; q3 = q4, {n, 4, 10^5}]; seq

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

A352089 Tribonacci-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the tribonacci numbers (A278038).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 13, 14, 18, 20, 21, 24, 26, 27, 28, 30, 33, 36, 39, 40, 44, 46, 48, 56, 60, 68, 69, 72, 75, 76, 80, 81, 82, 84, 87, 88, 90, 94, 96, 100, 108, 115, 116, 120, 126, 128, 129, 132, 135, 136, 138, 140, 149, 150, 156, 162, 168, 174, 176, 177, 180

Offset: 1

Amiram Eldar, Mar 04 2022

Numbers k such that A278043(k) | k.
The positive tribonacci numbers (A000073) are all terms.
If k = A000073(A042964(m)) is an odd tribonacci number, then k+1 is a term.
Ray (2005) and Ray and Cooper (2006) called these numbers "3-Zeckendorf Niven numbers" and proved that their asymptotic density is 0. - Amiram Eldar, Sep 06 2024

			6 is a term since its minimal tribonacci representation, A278038(6) = 110, has A278043(6) = 2 1's and 6 is divisible by 2.

Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

Cf. A000073, A042964, A278038, A278043.
Subsequences: A352090, A352091, A352092.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Divisible[n, DigitCount[Total[2^(s - 1)], 2, 1]]]; Select[Range[180], q]

A352107 Lazy-tribonacci-Niven numbers: numbers that are divisible by the number of terms in their maximal (or lazy) representation in terms of the tribonacci numbers (A352103).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 20, 21, 24, 28, 30, 33, 36, 39, 40, 48, 50, 56, 60, 68, 70, 72, 75, 76, 80, 90, 96, 100, 108, 115, 116, 120, 135, 136, 140, 150, 155, 156, 160, 162, 168, 175, 176, 177, 180, 184, 185, 188, 195, 198, 204, 205, 208, 215, 216, 225, 231, 260

Offset: 1

Amiram Eldar, Mar 05 2022

Numbers k such that A352104(k) | k.

			6 is a term since its maximal tribonacci representation, A352103(6) = 110, has A352104(6) = 2 1's and 6 is divisible by 2.

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

Cf. A352103, A352104.
Subsequences: A352108, A352109, A352110.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; q[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, False, Divisible[n, Total[v[[i[[1, 1]] ;; -1]]]]]]; Select[Range[300], q]

cp's OEIS Frontend

A334309 Numbers k such that k and k+1 are both base phi Niven numbers (A334308).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A334310 Starts of runs of 3 consecutive base phi Niven numbers (A334308).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A334311 Starts of runs of 4 consecutive base phi Niven numbers (A334308).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352089 Tribonacci-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the tribonacci numbers (A278038).

Views

Author

Keywords