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 10 results.

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]

A352108 Numbers k such that k and k+1 are both lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

1, 20, 39, 75, 115, 135, 155, 175, 176, 184, 204, 215, 264, 567, 684, 704, 725, 791, 846, 872, 1089, 1104, 1115, 1134, 1183, 1184, 1211, 1224, 1407, 1575, 1840, 1880, 2064, 2075, 2151, 2191, 2232, 2259, 2260, 2415, 2529, 2583, 2624, 2780, 2820, 2848, 2888, 2988
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2022

Keywords

Examples

			20 is a term since 20 and 21 are both lazy-tribonacci-Niven numbers: the maximal tribonacci representation of 20, A352103(20) = 10111, has 4 1's and 20 is divisible by 4, and the maximal tribonacci representation of 21, A352103(20) = 11001, has 3 1's and 21 is divisible by 3.

Links

Crossrefs

Cf. A352103, A352104.
Subsequence of A352107.
Subsequences: A352109, A352110.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090.

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[3000], q[#] && q[# + 1] &]

A352110 Starts of runs of 4 consecutive lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

1081455, 1976895, 2894175, 5886255, 6906912, 15604110, 16588752, 19291479, 20387232, 25919439, 32394942, 34801557, 35654175, 36813582, 36907899, 39117219, 41407392, 43520832, 46181055, 47954499, 52145952, 54524319, 54815397, 56733639, 57775102, 58942959, 59292177
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2022

Keywords

Comments

Conjecture: There are no runs of 5 consecutive lazy-tribonacci-Niven numbers (checked up to 6*10^9).

Examples

			1081455 is a term since 1081455, 1081456, 1081457 and 1081458 are all divisible by the number of terms in their maximal tribonacci representation:
        k               A352103(k)   A352104(k)    k/A352104(k)
  -------  -----------------------   ----------    ------------
  1081455  10101011011110110011110           15           72097
  1081456  10101011011110110011111           16           67591
  1081457  10101011011110110100100           13           83189
  1081458  10101011011110110100101           14           77247

Links

Crossrefs

Cf. A352103, A352104.
Subsequence of A352107, A352108 and A352109.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092.

A352322 Starts of runs of 3 consecutive Pell-Niven numbers (A352320).

Original entry on oeis.org

4, 28, 110, 168, 984, 1024, 3123, 3514, 5740, 6783, 6923, 8584, 12664, 16744, 18160, 19670, 23190, 23470, 24030, 34503, 34643, 36304, 40384, 45880, 47390, 50910, 51190, 51750, 57607, 61640, 68104, 73600, 78403, 78630, 78910, 79470, 86674, 89360, 95824, 101320
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Comments

Conjecture: There are no runs of 4 consecutive Pell-Niven numbers (checked up to 2*10^8).

Examples

			4 is a term since 4, 5 and 6 are all Pell-Niven numbers: the minimal Pell representation of 4, A317204(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, the minimal Pell representation of 5, A317204(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1, and the minimal Pell representation of 6, A317204(6) = 101, has the sum of digits 1+0+1 = 2 and 6 is divisible by 2.

Links

Crossrefs

Cf. A000129, A265744, A317204.
A182190 \ {0} is a subsequence.
Subsequence of A352320 and A352321.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109.

Programs

  • Mathematica
    pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellNivenQ[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]]]; seq[count_, nConsec_] := Module[{pn = pellNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ pn, c++; AppendTo[s, k - nConsec]]; pn = Join[Rest[pn], {pellNivenQ[k]}]; k++]; s]; seq[30, 3]

A352344 Starts of runs of 3 consecutive lazy-Pell-Niven numbers (A352342).

Original entry on oeis.org

2196, 2650, 5784, 17459, 28950, 57134, 112878, 124506, 147078, 162809, 169694, 191538, 210494, 218654, 223344, 223459, 230894, 239360, 258740, 277455, 278900, 285615, 289695, 291328, 291858, 295408, 311524, 314658, 324734, 332894, 335179, 341900, 347718, 362880
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Examples

			2196 is a term since 2196, 2197 and 2198 are all divisible by the sum of the digits in their maximal Pell representation:
     k  A352339(k)  A352340(k)  k/A352340(k)
  ----  ----------  ----------  ------------
  2196   121222020          12           183
  2197   121222021          13           169
  2198   121222022          14           157

Links

Crossrefs

Cf. A352339, A352340.
Subsequence of A352342 and A352343.
A352345 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322.

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]]; lazyPellNivenQ[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]]]]; seq[count, nConsec_] := Module[{lpn = lazyPellNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ lpn, c++; AppendTo[s, k - nConsec]]; lpn = Join[Rest[lpn], {lazyPellNivenQ[k]}]; k++]; s]; seq[30, 3]

A352510 Starts of runs of 3 consecutive Catalan-Niven numbers (A352508).

Original entry on oeis.org

4, 55, 144, 145, 511, 2943, 6950, 7734, 9470, 9750, 15630, 15631, 35034, 35464, 41590, 41986, 64735, 68523, 68870, 77510, 81150, 90958, 106063, 118264, 119043, 135970, 139403, 163188, 164862, 164863, 171346, 181510, 200759, 202761, 202762, 208024, 209230, 209586
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Examples

			4 is a term since 4, 5 and 6 are all Catalan-Niven numbers: the Catalan representation of 4, A014418(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, the Catalan representation of 5, A014418(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1, and the Catalan representation of 6, A014418(6) = 101, has the sum of digits 1+0+1 = 2 and 6 is divisible by 2.

Links

Crossrefs

Cf. A000108, A014418, A014420.
Subsequence of A352508 and A352509.
A352511 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352344.

Programs

  • Mathematica
    c[n_] := c[n] = CatalanNumber[n]; catNivQ[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]]]; seq[count_, nConsec_] := Module[{cn = catNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {catNivQ[k]}]; k++]; s]; seq[30, 3]

A364218 Starts of runs of 3 consecutive integers that are Jacobsthal-Niven numbers (A364216).

Original entry on oeis.org

1, 2, 14, 42, 43, 44, 86, 182, 544, 686, 846, 854, 1014, 1375, 1384, 1504, 1624, 2105, 2190, 2315, 2358, 2731, 2732, 2763, 2774, 2824, 3243, 3534, 3702, 4205, 4878, 5046, 5408, 5462, 5643, 5663, 6222, 6390, 6935, 7566, 7734, 7928, 8224, 8704, 8910, 9078, 9368
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2023

Keywords

Links

Crossrefs

Subsequence of A364216 and A364217.
Subsequence of A364219, A364220, A364221.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352344, A352510.

Programs

  • Mathematica
    consecJacobsthalNiven[10^4, 3] (* using the function from A364217 *)
  • PARI
    lista(10^4, 3) \\ using the function from A364217

A364381 Starts of runs of 3 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 14, 20, 26, 42, 43, 44, 84, 85, 86, 104, 115, 170, 182, 304, 344, 362, 414, 544, 682, 686, 692, 784, 854, 1014, 1370, 1384, 1504, 1673, 1685, 1706, 2224, 2315, 2358, 2730, 2731, 2732, 2763, 2774, 3243, 3594, 3702, 4144, 4688, 4864, 5046, 5408
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2023

Keywords

Links

Crossrefs

Cf. A265745, A265747.
Subsequence of A364379 and A364380.
Subsequences: A364382, A364383.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352344, A352510, A364218.

Programs

  • Mathematica
    consecGreedyJN[5500, 3] (* using the function consecGreedyJN from A364380 *)
  • PARI
    lista(5500, 3) \\ using the function lista from A364380

A364008 Starts of runs of 3 consecutive integers that are Wythoff-Niven numbers (A364006).

Original entry on oeis.org

6, 54, 374, 375, 978, 979, 14695, 15694, 17708, 17709, 34990, 36476, 38374, 41699, 45304, 75944, 85149, 93104, 113463, 114560, 116170, 117754, 120274, 121371, 203983, 221804, 250118, 259819, 270214, 270477, 275526, 276912, 288125, 297241, 297515, 299824, 309440
Offset: 1

Views

Author

Amiram Eldar, Jul 01 2023

Keywords

Links

Crossrefs

Subsequence of A364006 and A364007.
A364009 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352510.

Programs

  • Mathematica
    seq[10, 3] (* generates the first 10 terms using the function seq[count, nConsec] from A364007 *)

A364125 Starts of runs of 3 consecutive integers that are Stolarsky-Niven numbers (A364123).

Original entry on oeis.org

1419, 2680, 6984, 18765, 20383, 28390, 48697, 55560, 69056, 121913, 125340, 125341, 125739, 133614, 135189, 136409, 140789, 147563, 150138, 155518, 157068, 171819, 317933, 318188, 319395, 323685, 339723, 340846, 349326, 356290, 371041, 389010, 392903, 393809, 400608
Offset: 1

Views

Author

Amiram Eldar, Jul 07 2023

Keywords

Links

Crossrefs

Subsequence of A364123 and A364124.
Subsequence: A364126.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352344, A352510.

Programs

  • Mathematica
    seq[10, 3] (* generates the first 10 terms, using the function seq[count, nConsec] from A364124 *)
  • PARI
    lista(10, 3) \\ generates the first 10 terms, using the function lista(count, nConsec) from A364124
Showing 1-10 of 10 results.

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