cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 39 results. Next

A328209 Numbers m such that m and m+1 are consecutive Zeckendorf-Niven numbers (A328208).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 13, 21, 26, 55, 68, 80, 89, 92, 93, 110, 152, 183, 195, 207, 233, 236, 237, 254, 291, 304, 327, 364, 377, 380, 381, 398, 435, 471, 484, 555, 584, 605, 609, 639, 644, 759, 795, 834, 875, 894, 930, 987, 992, 1004, 1011, 1028, 1047, 1076, 1220
Offset: 1

Views

Author

Amiram Eldar, Oct 07 2019

Keywords

Examples

			12 is in the sequence since both 12 and 13 are in A328208: A007895(12) = 3 is a divisor of 12, and A007895(13) = 1 is a divisor of 13.

Links

Crossrefs

Cf. A005349, A007895, A328208.

Programs

  • Mathematica
    z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s (* after Alonso del Arte at A007895 *)

A328210 Starts of runs of 3 consecutive Zeckendorf-Niven numbers (A328208).

Original entry on oeis.org

1, 2, 3, 4, 12, 92, 236, 380, 1850, 2630, 4184, 7010, 8183, 8360, 11944, 12754, 13550, 16024, 17710, 17714, 18710, 20628, 22323, 22624, 25564, 28910, 31506, 36463, 36484, 39746, 40368, 44694, 48244, 49294, 53543, 58910, 59164, 64743, 70398, 75024, 77874, 78184
Offset: 1

Views

Author

Amiram Eldar, Oct 07 2019

Keywords

Examples

			12 is in the sequence since 12, 13 and 14 are in A328208: A007895(12) = 3 is a divisor of 12, A007895(13) = 1 is a divisor of 13, and A007895(14) = 2 is a divisor of 14.

Links

Crossrefs

Cf. A005349, A007895, A154701, A328208.

Programs

  • Mathematica
    z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {3}]; While[c < 50, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 2]]]; k++]; s (* after Alonso del Arte at A007895 *)

A328211 Starts of runs of 4 consecutive Zeckendorf-Niven numbers (A328208).

Original entry on oeis.org

1, 2, 3, 123543, 124242, 545502, 1367583, 1856349, 2431230, 2465110, 2593590, 2783709, 3247389, 3479229, 3917823, 3942909, 4174749, 4303428, 4494390, 4920640, 5143830, 5710383, 6261309, 6493149, 6552903, 6956829, 7420509, 7470880, 8970948, 9107790, 9507069, 10952928
Offset: 1

Views

Author

Amiram Eldar, Oct 07 2019

Keywords

Comments

Grundman proved that this sequence is infinite by showing the F(120k-6) + F(8) + F(6) + F(4) is a term for all k >= 1, where F(k) is the k-th Fibonacci number.
She also proved that the only starts of runs of 5 consecutive Zeckendorf-Niven numbers are 1 and 2.

Examples

			1 is in the sequence since 1, 2, 3 and 4 are in A328208: A007895(1) = 1 is a divisor of 1, A007895(2) = 1 is a divisor of 2, A007895(3) = 1 is a divisor of 3, and A007895(4) = 2 is a divisor of 4.

Links

Crossrefs

Cf. A005349, A007895, A141769, A328208.

Programs

  • Mathematica
    z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {4}]; While[c < 32, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 3]]]; k++]; s (* after Alonso del Arte at A007895 *)

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

Views

Author

Amiram Eldar, Dec 27 2019

Keywords

Examples

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

Links

Crossrefs

Intersection of A328208 and A328212.
Cf. A007895, A014417, A104326, A112310.

Programs

  • Mathematica
    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[#]] &]

A377209 Zeckendorf-Niven numbers (A328208) k such that k/z(k) is also a Zeckendorf-Niven number, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 21, 24, 26, 30, 34, 36, 42, 48, 55, 60, 66, 68, 72, 78, 81, 89, 90, 108, 110, 120, 126, 135, 144, 152, 168, 178, 180, 192, 204, 207, 233, 240, 243, 264, 270, 276, 288, 300, 304, 312, 324, 330, 336, 360, 377, 380, 390, 396, 408
Offset: 1

Views

Author

Amiram Eldar, Oct 20 2024

Keywords

Examples

			12 is a term since 12/z(12) = 4 is an integer and also 4/z(4) = 2 is an integer.

Links

Crossrefs

Cf. A007895, A376616 (binary analog).
Subsequence of A328208.
Subsequences: A000045, A377210.

Programs

  • Mathematica
    zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k]}, Divisible[k, z] && Divisible[k/z, zeck[k/z]]]; Select[Range[400], q]
  • PARI
    zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV at A007895
is(k) = {my(z = zeck(k)); !(k % z) && !((k/z) % zeck(k/z)); }

A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 21, 24, 26, 30, 34, 42, 48, 55, 60, 68, 78, 89, 110, 120, 126, 144, 178, 180, 192, 204, 233, 243, 264, 270, 288, 300, 312, 324, 330, 360, 377, 466, 480, 534, 540, 576, 600, 610, 621, 672, 720, 754, 768, 864, 987, 1020, 1056
Offset: 1

Views

Author

Amiram Eldar, Oct 20 2024

Keywords

Examples

			24 is a term since 24/z(24) = 12, 12/z(12) = 4 and 4/z(4) = 2 are all integers.

Links

Crossrefs

Cf. A000045 (a subsequence), A007895, A376617 (binary analog).
Subsequence of A328208 and A377209.

Programs

  • Mathematica
    zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k], z2, m, n}, IntegerQ[m = k/z] && Divisible[m, z2 = zeck[m]] && Divisible[n = m/z2, zeck[n]]]; Select[Range[1000], q]
  • PARI
    zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV at A007895
is(k) = {my(z = zeck(k), z2, m); if(k % z, return(0)); m = k/z; z2 = zeck(m); !(m % z2) && !((m/z2) % zeck(m/z2)); }

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

Views

Author

Amiram Eldar, Dec 27 2019

Keywords

Comments

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?

Examples

			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.

Links

Crossrefs

Intersection of A328209 and A328213.
Cf. A328208, A328210, A328212, A328214, A330711.

A376028 Zeckendorf-Niven numbers (A328208) with a record gap to the next Zeckendorf-Niven number.

Original entry on oeis.org

1, 6, 18, 30, 36, 48, 208, 5298, 6132, 6601, 8280, 12228, 17052, 68220, 113990, 120504, 438570, 1015416, 1343232, 1848400, 5338548, 12727143, 83877810, 330963120, 409185360, 418561770, 2428646640, 2834120595, 2876557200, 2940992640, 7218753758, 7306145012, 7609637140
Offset: 1

Views

Author

Amiram Eldar, Sep 06 2024

Keywords

Comments

The corresponding record gaps are 1, 2, 3, 4, 6, 7, 20, ... (see the link for more values).
Ray (2005) and Ray and Cooper (2006) proved that the asymptotic density of the Zeckendorf-Niven numbers is 0. Therefore, this sequence is infinite.

Examples

			6 is a term since it is a Zeckendorf-Niven number, and the next Zeckendorf-Niven number is 8, with a gap 8 - 6 = 2, which is a record since all the numbers below 6 are also Zeckendorf-Niven numbers.

References

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

Links

Crossrefs

Cf. A328208, A328209.
Similar sequences: A337076, A337077, A347495, A347496, A376029.

Programs

  • Mathematica
    z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; znQ[n_] := Divisible[n, z[n]]; seq[kmax_] := Module[{gapmax = 0, gap, k1 = 1, s = {}}, Do[If[znQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^4]

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

Views

Author

Amiram Eldar, Mar 20 2020

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

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

Views

Author

Amiram Eldar, Jan 27 2020

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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]
Showing 1-10 of 39 results. Next

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