A328212 -id:A328212 - cp's OEIS frontend

A328213 Numbers m such that m and m+1 are consecutive lazy-Fibonacci-Niven numbers (A328212).

1, 15, 35, 369, 549, 710, 890, 1089, 1989, 2196, 2629, 2951, 3311, 3679, 3684, 4927, 5268, 5291, 5488, 5603, 5775, 5819, 5928, 5940, 6524, 7072, 7364, 7403, 7475, 7500, 7657, 7812, 8567, 9254, 9360, 9372, 9383, 9477, 9841, 10205, 10304, 10360, 11816, 13299, 13454

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			15 is in the sequence since both 15 and 16 are in A328212: A112310(15) = 3 is a divisor of 15, and A112310(16) = 4 is a divisor of 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A112310, A328212.

ooQ[n_] := Module[{k = n}, While[k > 3, If[Divisible[k, 4], Return[True], k = Quotient[k, 2]]]; False]; c = 0; cn = 0; k = 1; s = {}; v = Table[-1, {2}]; While[cn < 45, If[! ooQ[k], c++; d = Total@IntegerDigits[k, 2]; If[Divisible[c, d], v = Join[Rest[v], {c}]; If[AllTrue[Differences[v], # == 1 &], cn++; AppendTo[s, c - 1]]]]; k++]; s

A328214 Starts of runs of 3 consecutive lazy-Fibonacci-Niven numbers (A328212).

Original entry on oeis.org

27312, 37504, 48060, 83248, 198254, 269856, 319694, 386136, 423520, 434300, 518175, 525672, 539800, 572184, 690858, 701118, 793799, 886534, 998866, 1015035, 1258444, 1396582, 1409058, 1511600, 1557422, 1680378, 1729398, 1753818, 2044768, 2136263, 2310624, 2396438, 2421024

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			27312 is in the sequence since 27312, 27313 and 27314 are in A328212: A112310(27312) = 16 is a divisor of 27312, A112310(27312) = 13 is a divisor of 27313, and A112310(27314) = 14 is a divisor of 27314.

Amiram Eldar, Table of n, a(n) for n = 1..79
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A112310, A154701, A328212.

ooQ[n_] := Module[{k = n}, While[k > 3, If[Divisible[k, 4], Return[True], k = Quotient[k, 2]]]; False]; c = 0; cn = 0; k = 1; s = {}; v = Table[-1, {3}]; While[cn < 33, If[! ooQ[k], c++; d = Total@IntegerDigits[k, 2]; If[Divisible[c, d], v = Join[Rest[v], {c}]; If[AllTrue[Differences[v], # == 1 &], cn++; AppendTo[s, c - 2]]]]; k++]; s

A328215 Starts of runs of 4 consecutive lazy-Fibonacci-Niven numbers (A328212).

Original entry on oeis.org

3674769, 17434975, 22711023, 26152125, 32784723, 41221725, 57846123, 93416568, 101681916, 122873490, 173504940, 225947148, 234209247, 259557450, 333681684, 377858544, 396241410, 413770056, 432640989, 443496447, 444571650, 484381323, 497625360, 556123167, 564869940

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

Grundman found a(1) and proved that there are no runs of 5 consecutive lazy-Fibonacci-Niven numbers.

			3674769 is in the sequence since 3674769, 3674770, 3674771 and 3674772 are in A328212: A112310(3674769) = 21 is a divisor of 3674769, A112310(3674770) = 22 is a divisor of 3674770, A112310(3674771) = 17 is a divisor of 3674771, and A112310(3674772) = 18 is a divisor of 3674772.

Amiram Eldar, Table of n, a(n) for n = 1..72
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A112310, A141769, A328212.

ooQ[n_] := Module[{k = n}, While[k > 3, If[Divisible[k, 4], Return[True], k = Quotient[k, 2]]]; False]; c = 0; cn = 0; k = 1; s = {}; v = Table[-1, {4}]; While[cn < 10, If[! ooQ[k], c++; d = Total@IntegerDigits[k, 2]; If[Divisible[c, d], v = Join[Rest[v], {c}]; If[AllTrue[Differences[v], # == 1 &], cn++; AppendTo[s, c - 3]]]]; k++]; s

More terms from Amiram Eldar, Oct 23 2019

A330711 Numbers that are both Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 30, 36, 48, 55, 60, 72, 78, 84, 90, 102, 105, 126, 144, 156, 168, 180, 184, 192, 208, 238, 240, 252, 264, 304, 315, 320, 322, 344, 360, 370, 378, 396, 430, 432, 488, 528, 536, 540, 576, 590, 605, 609, 621, 639, 648, 657, 660, 672, 680, 702

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

			6 is in the sequence since A007895(6) = 2 and A112310(6) = 3, and both 2 and 3 are divisors of 6.

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

Intersection of A328208 and A328212.

Cf. A007895, A014417, A104326, A112310.

zeckSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
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]]]]];
Select[Range[1000], Divisible[#, zeckSum[#]] && Divisible[#, dualZeckSum[#]] &]

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

Original entry on oeis.org

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.

A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

Numbers k for which A276086(k) is in A373852. - Antti Karttunen, Jun 22 2024

			1 is a term since A276150(1) = 1 divides 1;
2 is a term since A276150(2) = 1 divides 2;

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A005349, A049345, A049445, A064150, A064438, A064481, A118363, A235168, A276150, A328208, A328212, A373834 (characteristic function)
Cf. A276086, A333427, A333428, A341433, A347496, A358977, A373833, A373852.
Positions of 0's in A373832.

max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]

isA333426 = A373834; \\ Antti Karttunen, Jun 22 2024

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]

A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 13, 14, 18, 24, 26, 27, 30, 34, 36, 48, 55, 60, 64, 68, 69, 72, 78, 84, 86, 87, 88, 89, 90, 93, 94, 96, 99, 100, 102, 108, 110, 112, 116, 120, 140, 144, 150, 155, 156, 160, 172, 176, 177, 178, 180, 183, 184, 188, 192, 195, 196, 200, 204

Offset: 1

Amiram Eldar, Jan 08 2020

The k-th Fibonacci number is a term for all odd k, since its negaFibonacci representation is 1 followed by (k-1) zeros.

			4 is a term since the negaFibonacci representation of 4 is 10010 whose sum of digits is 1 + 0 + 0 + 1 + 0 = 2 which is a divisor of 4.

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

Cf. A005349, A215022, A328208, A328212, A331083.

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[200], Divisible[#, negaFibTermsNum[#]] &]

A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

Original entry on oeis.org

1, 2, 6, 12, 15, 16, 18, 20, 30, 35, 36, 45, 48, 55, 60, 70, 72, 78, 84, 90, 91, 95, 96, 98, 104, 108, 132, 144, 147, 154, 168, 175, 184, 189, 208, 224, 231, 232, 245, 252, 256, 261, 264, 270, 275, 280, 282, 287, 294, 315, 322, 324, 330, 336, 340, 342, 351, 357

Offset: 1

Amiram Eldar, Apr 22 2020

			6 is a term since its base phi representation is 1010.0001, and the number of 1's is 3, which is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Wikipedia, Golden ratio base.

Cf. A005349, A049445, A055778, A064150, A064438, A064481, A118363, A328208, A328212, A333426.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]]; Select[Range[360], Divisible[#, phiDigSum[#]] &]

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]

cp's OEIS Frontend

A328213 Numbers m such that m and m+1 are consecutive lazy-Fibonacci-Niven numbers (A328212).

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A328214 Starts of runs of 3 consecutive lazy-Fibonacci-Niven numbers (A328212).

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A328215 Starts of runs of 4 consecutive lazy-Fibonacci-Niven numbers (A328212).

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A330711 Numbers that are both Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

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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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A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

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

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A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

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A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

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