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.

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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

Views

Author

Amiram Eldar, Mar 04 2022

Keywords

Comments

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

Examples

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

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. 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.

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Mar 05 2022

Keywords

Comments

Numbers k such that A352104(k) | k.

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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]

A352320 Pell-Niven numbers: numbers that are divisible by the sum of the digits in their minimal (or greedy) representation in terms of the Pell numbers (A317204).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 14, 15, 18, 20, 24, 28, 29, 30, 33, 34, 36, 39, 40, 42, 44, 48, 50, 58, 60, 63, 64, 68, 70, 72, 82, 84, 87, 88, 90, 92, 96, 110, 111, 112, 115, 116, 120, 125, 126, 135, 140, 141, 144, 155, 164, 165, 168, 169, 170, 174, 180, 183, 184, 186
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Comments

Numbers k such that A265744(k) | k.
All the positive Pell numbers (A000129) are terms.

Examples

			6 is a term since its minimal Pell representation, A317204(6) = 101, has A265744(6) = 2 1's and 6 is divisible by 2.

Links

Crossrefs

Cf. A000129, A265744, A317204.
Subsequences: A352321, A352322.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107.

Programs

  • Mathematica
    pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Divisible[n, Plus @@ IntegerDigits[ Total[3^(s - 1)], 3]]]; Select[Range[200], q]

A352342 Lazy-Pell-Niven numbers: numbers that are divisible by the sum of the digits in their maximal (or lazy) representation in terms of the Pell numbers (A352339).

Original entry on oeis.org

1, 2, 4, 9, 12, 15, 20, 24, 25, 28, 30, 35, 40, 48, 50, 54, 56, 60, 63, 64, 70, 72, 78, 84, 88, 91, 96, 102, 115, 120, 136, 144, 160, 162, 168, 180, 182, 184, 189, 207, 209, 210, 216, 217, 234, 246, 256, 261, 270, 304, 306, 308, 315, 320, 328, 333, 350, 352, 357
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Comments

Numbers k such that A352340(k) | k.

Examples

			4 is a term since its maximal Pell representation, A352339(4) = 11, has the sum of digits A352340(4) = 1+1 = 2 and 4 is divisible by 2.

Links

Crossrefs

Cf. A352339, A352340.
Subsequences: A352343, A352344, A352345.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320.

Programs

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

A352508 Catalan-Niven numbers: numbers that are divisible by the sum of the digits in their representation in terms of the Catalan numbers (A014418).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 14, 16, 18, 21, 24, 28, 30, 32, 33, 40, 42, 44, 45, 48, 55, 56, 57, 60, 65, 72, 78, 80, 84, 88, 95, 100, 105, 112, 126, 128, 130, 132, 134, 135, 138, 140, 144, 145, 146, 147, 152, 155, 156, 168, 170, 174, 180, 184, 185, 195, 210, 216
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Comments

Numbers k such that A014420(k) | k.
All the Catalan numbers (A000108) are terms.
If k is an odd Catalan number (A038003), then k+1 is a term.

Examples

			4 is a term since its Catalan representation, A014418(4) = 20, has the sum of digits A014420(4) = 2 + 0 = 2 and 4 is divisible by 2.
9 is a term since its Catalan representation, A014418(9) = 120, has the sum of digits A014420(9) = 1 + 2 + 0 = 3 and 9 is divisible by 3.

Links

Crossrefs

Cf. A014418, A014420.
Subsequences: A000108, A038003, A352509, A352510, A352511.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342.

Programs

  • Mathematica
    c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[216], q]

A331088 Positive numbers k such that -k is a negative negaFibonacci-Niven number, i.e., divisible by the number of terms in its negaFibonacci representation (A331084).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 18, 20, 21, 22, 24, 27, 30, 36, 42, 44, 45, 48, 50, 51, 54, 55, 56, 57, 58, 60, 66, 72, 75, 76, 80, 84, 90, 92, 96, 100, 104, 105, 108, 110, 111, 112, 115, 116, 120, 124, 126, 128, 129, 132, 136, 138, 141, 142, 144, 150, 152, 153, 156, 168, 170, 172, 175, 176, 180, 184, 186, 190, 192, 196, 198
Offset: 1

Views

Author

Amiram Eldar, Jan 08 2020

Keywords

Comments

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

Examples

			4 is a term since the negaFibonacci representation of -4 is 1010 whose sum of digits is 1 + 0 + 1 + 0 = 2 which is a divisor of 4.

Links

Crossrefs

Cf. A005349, A215023, A328208, A328212, A331083, A331084, A331085.

Programs

  • Mathematica
    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[-#]] &]

A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 16, 20, 22, 24, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 68, 72, 75, 76, 84, 86, 87, 88, 92, 93, 95, 96, 99, 100, 104, 105, 108, 112, 115, 117, 120, 125, 126, 128, 129, 132, 135, 136, 138, 140
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2023

Keywords

Comments

Numbers k such that A364215(k) | k.
A007583 is a subsequence since A364215(A007583(n)) = 1 for n >= 0.

Links

Crossrefs

Cf. A280049, A364215.
Subsequences: A007583, A364217, A364218, A364219, A364220, A364221.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

Programs

  • Mathematica
    seq[kmax_] := Module[{m = 1, s = {}}, Do[If[Divisible[k, DigitCount[m, 2, 1]], AppendTo[s, k]]; While[m++; OddQ[IntegerExponent[m, 2]]], {k, 1, kmax}]; s]; seq[140]
  • PARI
    lista(kmax) = {my(m = 1); for(k = 1, kmax, if( !(k % sumdigits(m, 2)), print1(k,", ")); until(valuation(m, 2)%2 == 0, m++));}

A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 20, 21, 22, 24, 26, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 64, 68, 69, 72, 75, 76, 80, 84, 85, 86, 87, 88, 90, 92, 93, 96, 99, 100, 104, 105, 106, 108, 111, 112, 115, 116, 117, 120
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2023

Keywords

Comments

Numbers k such that A265745(k) | k.
The positive Jacobsthal numbers, A001045(n) for n >= 1, are terms since their representation in Jacobsthal greedy base is one 1 followed by n-1 0's, so A265745(A001045(n)) = 1 divides A001045(n).

Links

Crossrefs

Cf. A265745, A265747.
Subsequences: A364380, A364381, A364382, A364383.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508, A364216.

Programs

  • Mathematica
    greedyJacobNivenQ[n_] := Divisible[n, A265745[n]]; Select[Range[120], greedyJacobNivenQ] (* using A265745[n] *)
  • PARI
    isA364379(n) = !(n % A265745(n)); \\ using A265745(n)

A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 33, 34, 36, 39, 40, 42, 44, 48, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 77, 78, 80, 84, 90, 92, 96, 100, 102, 104, 108, 111, 112, 114, 115, 116, 120, 123, 124, 126, 128, 129, 130, 132, 135, 136, 138, 140
Offset: 1

Views

Author

Amiram Eldar, Jan 27 2020

Keywords

Examples

			6 is a term since A039724(-6) = 1110 and 1 + 1 + 1 + 0 = 3 is a divisor of 6.

Links

Crossrefs

Cf. A005352, A027615, A039724, A005349, A049445, A328208, A328212, A331085, A331088, A331728.

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]

A364006 Wythoff-Niven numbers: numbers that are divisible by the number of 1's in their Wythoff representation.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 12, 15, 18, 20, 21, 24, 26, 28, 32, 35, 39, 40, 42, 45, 47, 51, 52, 54, 55, 56, 60, 68, 72, 76, 80, 84, 86, 88, 90, 91, 98, 100, 102, 105, 117, 120, 123, 125, 135, 136, 138, 141, 143, 144, 156, 164, 168, 172, 174, 176, 178, 180, 188, 192
Offset: 1

Views

Author

Amiram Eldar, Jul 01 2023

Keywords

Comments

Numbers k such that A135818(k) | k.
Includes all the positive even-indexed Fibonacci numbers (A001906), since the Wythoff representation of Fibonacci(2*n), for n >= 1, is 1 followed by n-1 0's.

Links

Crossrefs

Cf. A135818, A189921.
Subsequences: A364007, A364008, A364009.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352508.

Programs

  • Mathematica
    wnQ[n_] := (s = Total[w[n]]) > 0 && Divisible[n, s] (* using the function w[n] from A364005 *)
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