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 1-7 of 7 results.

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]

A352509 Numbers k such that k and k+1 are both Catalan-Niven numbers (A352508).

Original entry on oeis.org

1, 4, 5, 9, 32, 44, 55, 56, 134, 144, 145, 146, 155, 184, 234, 324, 329, 414, 426, 429, 434, 455, 511, 512, 603, 636, 930, 1004, 1014, 1160, 1183, 1215, 1287, 1308, 1448, 1472, 1505, 1562, 1595, 1808, 1854, 1967, 1985, 1995, 2051, 2075, 2096, 2135, 2165, 2255
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Examples

			4 is a term since 4 and 5 are both 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, and the Catalan representation of 5, A014418(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1.

Links

Crossrefs

Cf. A000108, A014418, A014420.
Subsequence of A352508.
Subsequences: A038003, A352510, A352511.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343.

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[2300], q[#] && q[#+1] &]

A352511 Starts of runs of 4 consecutive Catalan-Niven numbers (A352508).

Original entry on oeis.org

144, 15630, 164862, 202761, 373788, 450189, 753183, 1403961, 1779105, 2588415, 2673774, 2814229, 2850880, 3009174, 3013722, 3045870, 3091023, 3702390, 3942519, 4042950, 4432128, 4725432, 4938348, 5718942, 5907312, 6268248, 6519615, 6592752, 6791379, 7095492, 8567802
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Comments

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

Examples

			144 is a term since 144, 145, 146 and 147 are all divisible by the sum of the digits in their Catalan representation:
    k  A014418(k)  A014420(k)  k/A014420(k)
  ---  ----------  ----------  ------------
  144      100210           4            36
  145      100211           5            29
  146      101000           2            73
  147      101001           3            49

Links

Crossrefs

Cf. A000108, A014418, A014420.
Subsequence of A352508, A352509 and A352510.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092, A352110, A352345.

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[5, 4]

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 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.