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 14 results. Next

A342428 Starts of runs of 3 consecutive Niven numbers in base 3/2 (A342426).

Original entry on oeis.org

2196, 7656, 15624, 16335, 64375, 109224, 171624, 202824, 328887, 329427, 392733, 393640, 447578, 482238, 494450, 520695, 631824, 723519, 773790, 785695, 820960, 876987, 981783, 986607, 1021824, 1026750, 1030455, 1084048, 1108094, 1160670, 1235070, 1242824, 1412908
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2021

Keywords

Examples

			2196 is a term since 2196, 2197 and 2198 are all Niven numbers in base 3/2.

Links

Crossrefs

Subsequence of A342426 and A342427.
Subsequences: A342429.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary).

Programs

  • Mathematica
    s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[10^6], AllTrue[# + {0, 1, 2}, q] &]

A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 2, 6, 7, 14, 30, 31, 62, 126, 127, 174, 184, 234, 243, 254, 304, 474, 483, 510, 511, 534, 543, 544, 783, 784, 903, 904, 954, 963, 1022, 1134, 1144, 1253, 1264, 1448, 1475, 1504, 1895, 1914, 1923, 1974, 2046, 2047, 2093, 2094, 2104, 2814, 2888, 2944, 3054, 3064
Offset: 1

Views

Author

Amiram Eldar, May 15 2021

Keywords

Examples

			1 is a term since 1, 2 and 3 are all Gray-code Niven numbers.

Links

Crossrefs

Cf. A005811, A014550.
Subsequence of A344341 and A344342.
Subsequences: A344344.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary), A342428 (base 3/2).

Programs

  • Mathematica
    gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[3000], AllTrue[# + {0, 1, 2}, gcNivenQ] &]

A351721 Starts of runs of 3 consecutive lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

607068, 640618, 665720, 900921, 1000880, 1375940, 1505878, 1537250, 1924224, 1938508, 1966338, 2527998, 3394224, 3935424, 4242624, 4476624, 4747224, 4794624, 5351367, 5401824, 5526024, 5636356, 5992298, 6103900, 6343298, 7028362, 7113024, 8879424, 8998262, 9431424
Offset: 1

Views

Author

Amiram Eldar, Feb 17 2022

Keywords

Comments

Conjecture: There are no runs of 4 consecutive lazy-Lucas-Niven numbers (checked up to 3*10^9).

Examples

			607068 is a term since 607068, 607069 and 607070 are all divisible by the number of terms in their maximal representation:
     k                   A130311(k)  A131343(k)  k/A131343(k)
-------------------------------------------------------------
607068  111010101010101011110111101         18          33726
607069  111010101010101011110111111         19          31951
607070  111010101010101011111010110         17          35710

Links

Crossrefs

Cf. A000032, A130311, A131343.
Subsequence of A351719 and A351720.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716.

A351716 Starts of runs of 3 consecutive Lucas-Niven numbers (A351714).

Original entry on oeis.org

1, 2, 6, 10, 1070, 4214, 10654, 10730, 13118, 31143, 39830, 43864, 47663, 48184, 50134, 62334, 63510, 79954, 83344, 84006, 89614, 107270, 119224, 119434, 121384, 124586, 124984, 129094, 129843, 148910, 165430, 167760, 168574, 183274, 193144, 198184, 198904, 199870
Offset: 1

Views

Author

Amiram Eldar, Feb 17 2022

Keywords

Comments

Conjecture: 1 is the only start of a run of 4 consecutive Lucas-Niven numbers (checked up to 10^9).

Examples

			6 is a term since 6, 7 and 8 are all Lucas-Niven numbers: the minimal Lucas representation of 6, A130310(6) = 1001, has 2 1's and 6 is divisible by 2, the minimal Lucas representation of 7, A130310(7) = 10000, has one 1 and 7 is divisible by 1, and the minimal Lucas representation of 8, A130310(8) = 10010, has 2 1's and 8 is divisible by 2.

Links

Crossrefs

Subsequence of A351714 and A351715.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351721.

Programs

  • Mathematica
    lucasNivenQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Divisible[n, Plus @@ IntegerDigits[Total[2^s], 2]]]; seq[count_, nConsec_] := Module[{luc = lucasNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ luc, c++; AppendTo[s, k - nConsec]]; luc = Join[Rest[luc], {lucasNivenQ[k]}]; k++]; s]; seq[50, 3]

A352091 Starts of runs of 3 consecutive tribonacci-Niven numbers (A352089).

Original entry on oeis.org

6, 12, 26, 80, 184, 506, 664, 1602, 1603, 1704, 3409, 6034, 9830, 15723, 16744, 19088, 21230, 21664, 22834, 33544, 39424, 40662, 40730, 51190, 55744, 56224, 60710, 61264, 63734, 66014, 66055, 67144, 67248, 73024, 78064, 81150, 84790, 94086, 95094, 109087, 111880
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2022

Keywords

Examples

			6 is a term since 6, 7 and 8 are all tribonacci-Niven numbers: the minimal tribonacci representation of 6, A278038(6) = 110, has 2 1's and 6 is divisible by 2, the minimal tribonacci representation of 7, A278038(7) = 1000, has one 1 and 7 is divisible by 1, and the minimal tribonacci representation of 8, A278038(8) = 1001, has 2 1's and 8 is divisible by 2.

Links

Crossrefs

Cf. A278038.
Subsequence of A352089 and A352090.
A352092 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721.

Programs

  • Mathematica
    t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; triboNivenQ[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]]]; seq[count_, nConsec_] := Module[{tri = triboNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {triboNivenQ[k]}]; k++]; s]; seq[30, 3]

A334311 Starts of runs of 4 consecutive base phi Niven numbers (A334308).

Original entry on oeis.org

285129, 1958893, 2501533, 6488440, 7069840, 8803023, 16514327, 23826399, 34031773, 52256248, 68198847, 72969138, 76779087, 77622950, 87430210, 87474672, 96485487, 114137958, 120197293, 136275022, 151444458, 173740578, 174878352, 183872325, 188385855, 196268415
Offset: 1

Views

Author

Amiram Eldar, Apr 22 2020

Keywords

Examples

			285129 is a term since 285129, 285130, 285131 and 285132 are all base phi Niven numbers.

Links

Crossrefs

Cf. A141769, A330933, A331824, A331825, A334308, A334309, A334310.

Programs

  • Mathematica
    phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; q1 = phiNivenQ[1]; q2 = phiNivenQ[2]; q3 = phiNivenQ[3]; seq = {}; Do[q4 = phiNivenQ[n]; If[q1 && q2 && q3 && q4, AppendTo[seq, n - 3]]; q1 = q2; q2 = q3; q3 = q4, {n, 4, 10^5}]; seq

A352109 Starts of runs of 3 consecutive lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

175, 1183, 2259, 5290, 12969, 21130, 51820, 70629, 78090, 79540, 81818, 129648, 160224, 169234, 180908, 228240, 238574, 249494, 278628, 332891, 376335, 383866, 398650, 399644, 454090, 550380, 565200, 683448, 683604, 694274, 728895, 754390, 782110, 809830, 837550
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2022

Keywords

Examples

			175 is a term since 175, 176 and 177 are all divisible by the number of terms in their maximal tribonacci representation:
    k  A352103(k)  A352104(k)  k/A352104(k)
  ---  ----------  ----------  ------------
  175    11111110           7            25
  176    11111111           8            22
  177   100100100           3            59

Links

Crossrefs

Cf. A352103, A352104.
Subsequence of A352107 and A352108.
A352110 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091.

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

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]
Showing 1-10 of 14 results. Next

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