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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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]

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]

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

A381583 Starts of runs of 3 consecutive integers that are all terms in A381581.

Original entry on oeis.org

1, 2, 20, 55, 56, 110, 304, 364, 398, 846, 1024, 1084, 1744, 1854, 2044, 2104, 2105, 2527, 2824, 2862, 3870, 4374, 5222, 5223, 5243, 5718, 5928, 6488, 6784, 6844, 6894, 6978, 7142, 7924, 10590, 11240, 11889, 11975, 12248, 14284, 14915, 16638, 17710, 17714, 17824
Offset: 1

Views

Author

Amiram Eldar, Feb 28 2025

Keywords

Comments

If k is congruent to 1 or 5 mod 12 (A087445), then A001906(k) = Fibonacci(2*k) is a term.

Examples

			1 is a term since A291711(1) = 1 divides 1, A291711(2) = 2 divides 2, and A291711(3) = 1 divides 3.
20 is a term since A291711(20) = 4 divides 20, A291711(21) = 1 divides 21, and A291711(22) = 2 divides 22.

Links

Crossrefs

Cf. A000045, A001906, A087445, A291711.
Subsequence of A381581 and A381582.
Subsequences: A381584, A381585.
Similar sequences: A154701, A328210, A330932, A351721.

Programs

  • Mathematica
    f[n_] := f[n] = Fibonacci[2*n]; q[n_] := q[n] = Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; seq[count_, nConsec_] := Module[{cn = q /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {q[k]}]; k++]; s]; seq[45, 3]
  • PARI
    mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
is1(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}
list(lim) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, lim, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

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 *)
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