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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A297242 Total variation of base-14 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 6
Offset: 1

Views

Author

Clark Kimberling, Jan 17 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers:

Examples

			2^20 in base 14:  1, 13, 4, 1, 12, 4; here, DV = 20 and UV = 23, so that a(2^20) = 43.

Links

Crossrefs

Cf. A297240, A297241, A297330.

Programs

  • Mathematica
    b = 14; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, e.g. A037834 *)

A297243 Down-variation of the base-15 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling, Jan 17 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.

Examples

			30 in base 15: 2,0; here DV = 2, so that a(30) = 2.

Links

Crossrefs

Cf. A297244, A297245, A297330.

Programs

  • Mathematica
    g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 15; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297243 *)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297244 *)

A297244 Up-variation of the base-15 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 0, 0, 0, 0, 1, 2, 3
Offset: 1

Views

Author

Clark Kimberling, Jan 17 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.

Examples

			19 in base 15: 1,4; here UV = 3, so that a(19) = 3.

Links

Crossrefs

Cf. A297243, A297245, A297330.

Programs

  • Mathematica
    g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 15; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297243 *)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297244 *)

A297245 Total variation of base-15 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 4, 3, 2, 1, 0, 1, 2, 3
Offset: 1

Views

Author

Clark Kimberling, Jan 17 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers:

Examples

			2^20 in base 15:  1, 5, 10, 10, 5, 1; here, DV = 9 and UV = 9, so that a(2^20) = 18.

Links

Crossrefs

Cf. A297243, A297244, A297330.

Programs

  • Mathematica
    b = 15; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, e.g. A037834 *)

A297246 Down-variation of the base-16 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.

Examples

			32 in base 15: 2,0; here DV = 2, so that a(32) = 2.

Links

Crossrefs

Cf. A297247, A297248, A297330.

Programs

  • Mathematica
    g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 16; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297246 *)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297247 *)

A297247 Up-variation of the base-16 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.

Examples

			20 in base 16: 1,4; here UV = 3, so that a(20) = 3.

Links

Crossrefs

Cf. A297246, A297248, A297330.

Programs

  • Mathematica
    g[n_, b_] := Differences[IntegerDigits[n, b]];
b = 16; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297246 *)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297247 *)

A297249 Numbers whose base-3 digits have greater down-variation than up-variation; see Comments.

Original entry on oeis.org

3, 6, 7, 9, 12, 15, 18, 19, 21, 22, 24, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Examples

			147 in base-3:  1,3,1,1,0, having DV = 3, UV = 2, so that 147 is in the sequence.

Links

Crossrefs

Cf. A297250, A297251, A297330.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 3; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297249 *)
Take[Flatten[Position[w, 0]], 120]    (* A297250 *)
Take[Flatten[Position[w, 1]], 120]    (* A297251 *)

A297251 Numbers whose base-3 digits have greater up-variation than down-variation; see Comments.

Original entry on oeis.org

5, 11, 14, 17, 29, 32, 35, 38, 41, 44, 47, 50, 53, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 275, 278, 281, 284, 287, 290
Offset: 1

Views

Author

Clark Kimberling, Apr 10 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Examples

			290 in base-3:  1,0,1,2,0,2, having DV = 3, UV = 4, so that 147 is in the sequence.

Links

Crossrefs

Cf. A297249, A297250, A297330.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 3; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297249 *)
Take[Flatten[Position[w, 0]], 120]    (* A297250 *)
Take[Flatten[Position[w, 1]], 120]    (* A297251 *)

A297252 Numbers whose base-4 digits have greater down-variation than up-variation; see Comments.

Original entry on oeis.org

4, 8, 9, 12, 13, 14, 16, 20, 24, 28, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 129, 132, 133, 136, 137, 140, 141, 144, 145, 148, 149, 152, 153
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Examples

			153 in base-4:  2,1,2,1, having DV = 2, UV = 1, so that 153 is in the sequence.

Links

Crossrefs

Cf. A297330, A297253, A297254.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 4; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297252 *)
Take[Flatten[Position[w, 0]], 120]    (* A297253 *)
Take[Flatten[Position[w, 1]], 120]    (* A297254 *)

A297253 Numbers whose base-4 digits having equal up-variation and down-variation; see Comments.

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 195, 199, 203, 207, 211, 215, 219, 223
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Examples

			223 in base-4:  3,2,3,3, having DV = 1, UV = 1, so that 223 is in the sequence.

Links

Crossrefs

Cf. A297330, A297252, A297254.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 4; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297252 *)
Take[Flatten[Position[w, 0]], 120]    (* A297253 *)
Take[Flatten[Position[w, 1]], 120]    (* A297254 *)
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