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.

A352343 Numbers k such that k and k+1 are both lazy-Pell-Niven numbers (A352342).

Original entry on oeis.org

1, 24, 63, 209, 216, 459, 560, 584, 656, 729, 999, 1110, 1269, 1728, 1859, 1989, 2100, 2196, 2197, 2255, 2650, 2651, 2820, 3443, 3497, 4080, 4563, 5291, 5784, 5785, 5837, 5928, 6252, 6383, 7344, 7657, 7812, 8150, 8203, 8459, 8670, 8749, 9251, 9295, 9372, 9464, 9840, 9884
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Comments

Numbers k such that A352340(k) | k and A352340(k+1) | k+1.

Examples

			24 is a term since 24 and 25 are both lazy-Pell-Niven numbers: the maximal Pell representation of 24, A352339(24) = 1210, has the sum of digits A352340(24) = 1+2+1+0 = 4 and 24 is divisible by 4, and the maximal Pell representation of 25, A352339(25) = 1211, has the sum of digits A352340(25) = 1+2+1+1 = 5 and 25 is divisible by 5.

Links

Crossrefs

Cf. A352339, A352340.
Subsequence of A352342.
Subsequences: A352344, A352345.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321.

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]]]]; Select[Range[10^4], lazyPellNivenQ[#] && lazyPellNivenQ[#+1] &]

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]

A352345 Starts of runs of 4 consecutive lazy-Pell-Niven numbers (A352342).

Original entry on oeis.org

750139, 41765247, 54831951, 56423275, 136038447, 151175724, 223956843, 227483124, 293913170, 362557214, 382572475, 457616575, 502106253, 562407324, 586380624, 637133390, 724382239, 771849439, 774421478, 859463253, 926398647, 953750523, 1043787390, 1193063550
Offset: 1

Views

Author

Amiram Eldar, Mar 12 2022

Keywords

Comments

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

Examples

			750139 is a term since 750139, 750140, 750141 and 750142 are all divisible by the sum of the digits in their maximal Pell representation:
       k        A352339(k)  A352340(k)  k/A352340(k)
  ------  ----------------  ---------   -----------
  750139  1102022021112220         19         39481
  750140  1102022021112221         20         37507
  750141  1102022021112222         21         35721
  750142  1102022021120210         17         44126

Crossrefs

Cf. A352339, A352340.
Subsequence of A352342, A352343 and A352344.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092, A352110.

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]

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)

A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 14, 16, 20, 22, 24, 27, 30, 36, 38, 40, 42, 44, 48, 54, 56, 57, 60, 65, 69, 72, 75, 80, 84, 85, 90, 92, 96, 98, 100, 102, 104, 108, 112, 116, 120, 124, 126, 132, 136, 138, 145, 147, 150, 153, 155, 159, 160, 175, 180, 185, 190, 195, 196, 205
Offset: 1

Views

Author

Amiram Eldar, Jul 07 2023

Keywords

Comments

Numbers k such that A200649(k) | k.
Fibonacci(k) + 1 is a term if k !== 3 (mod 6) (i.e., k is in A047263).

Examples

			4 is a term since its Stolarsky representation, A364121(4) = 10, has one 1 and 4 is divisible by 1.
6 is a term since its Stolarsky representation, A364121(6) = 101, has 2 1's and 6 is divisible by 2.

Links

Crossrefs

Cf. A047263, A200649, A364121.
Subsequences: A364124, A364125, A364126.
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
    stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolNivQ[n_] := n > 1 && Divisible[n, Total[stol[n]]];
Select[Range[200], stolNivQ]
  • PARI
    stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
isA364123(n) = n > 1 && !(n % vecsum(stol(n)));
Showing 1-7 of 7 results.

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

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