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 115 results. Next

A296882 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(pits) = #(peaks); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 1

Views

Author

Clark Kimberling, Jan 10 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296882-A296883 partition the natural numbers. See the guides at A296712. We have a(n) = A000027(n) for n=1..100 but not n=101.
.
Guide to related sequences:
Base #(pits) = #(peaks) #(pits) > #(peaks) #(pits) < #(peaks)
2 A296858 A296859 A296860
3 A296861 A296862 A296863
4 A296864 A296865 A296866
5 A296867 A296868 A296869
6 A296870 A296871 A296872
7 A296873 A296874 A296875
8 A296876 A296877 A296878
9 A296879 A296880 A296881
10 A296882 A296883 A296884
11 A296885 A296886 A296887
12 A296888 A296889 A296890
13 A296891 A296892 A296893
14 A296894 A296895 A296896
15 A296897 A296898 A296899
16 A296900 A296901 A296902
20 A296903 A296904 A296905
60 A296906 A296907 A296908

Examples

			The base-10 digits of 1212 are 1,2,1,2; here #(pits) = 1 and #(peaks) = 1, so 1212 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296883, A296884.

Programs

  • Mathematica
    z = 200; b = 10;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296882 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296883 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296884 *)

Extensions

Overview table corrected by Georg Fischer, Aug 24 2021

A297030 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-2 digits of n; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 3, 3, 3, 3, 3, 2, 1, 2, 3, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 2, 1, 2, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 3, 3, 4, 5, 5, 5, 5, 5, 4, 3, 4, 4, 4, 3, 3, 2, 1, 2, 3, 4, 4, 5, 5, 5, 4, 5, 6, 6, 6, 6, 6, 5, 4, 4, 5, 6, 6, 6, 6, 6
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2018

Keywords

Comments

The definition of "piece" starts with the base-b digits d(m), d(m-1), ..., d(0) of n. First, an *ascent* is a list (d(i), d(i-1), ..., d(i-h)) such that d(i) < d(i-1) < ... < d(i-h), where d(i+1) >= d(i) if i < m, and d(i-h-1) >= d(i-h) if i > h. A *descent* is a list (d(i), d(i-1), ..., d(i-h)) such that d(i) > d(i-1) > ... > d(i-h), where d(i+1) <= d(i) if i < m, and d(i-h-1) <= d(i-h) if i > h. A *flat* is a list (d(i), d(i-1), ..., d(i-h)), where h > 0, such that d(i) = d(i-1) = ... = d(i-h), where d(i+1) != d(i) if i < m, and d(i-h-1) != d(i-h) if i > h. A *piece* is an ascent, a descent, or a flat. Example: 235621103 has five pieces: (2,3,5,6), (6,2,1), (1,1), (1,0), and (0,3); that's 2 ascents, 2 descents, and 1 flat. For every b, the "piece sequence" includes every positive integer infinitely many times.

Examples

			Base-2 digits for 100:  1, 1, 0, 0, 1, 0, 0, so that a(100) = 6.

Links

Crossrefs

Cf. A297038, A296712 (rises and falls), A296882 (pits and peaks).
Guide to related sequences:
Base # pieces for n >= 1
2 A297030
3 A297031
4 A297032
5 A297033
6 A297034
7 A297035
8 A297036
9 A297037
10 A297038
11 A297039
12 A297040
13 A297041
14 A297042
15 A297043
16 A297044
20 A297045
60 A297046

Programs

  • Mathematica
    a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
b = 2; Table[a[n, b], {n, 1, 120}]

A296894 Numbers whose base-14 digits d(m), d(m-1), ..., d(0) have #(pits) = #(peaks); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296894-A296897 partition the natural numbers. See the guides at A296712 and A296882.

Examples

			The base-14 digits of 3152 are 1,2,1,2; here #(pits) = 1 and #(peaks) = 1, so 3152 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296895, A296896.

Programs

  • Mathematica
    z = 200; b = 14;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296894 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296895 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296896 *)

A296762 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(rises) = #(falls); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 421
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2018

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296762-A296764 partition the natural numbers. See the guide at A296712.

Examples

			The base-20 digits of 421 are 1,1,1; here #(rises) = 0 and #(falls) = 0, so 421 is in the sequence.

Links

Crossrefs

Cf. A296763, A296764, A296712.

Programs

  • Mathematica
    z = 200; b = 20; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296762 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296763 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296764 *)

A296764 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(rises) < #(falls); see Comments.

Original entry on oeis.org

20, 40, 41, 60, 61, 62, 80, 81, 82, 83, 100, 101, 102, 103, 104, 120, 121, 122, 123, 124, 125, 140, 141, 142, 143, 144, 145, 146, 160, 161, 162, 163, 164, 165, 166, 167, 180, 181, 182, 183, 184, 185, 186, 187, 188, 200, 201, 202, 203, 204, 205, 206, 207, 208
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2018

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296762-A296764 partition the natural numbers. See the guide at A296712.

Examples

			The base-20 digits of 208 are 10,8; here #(rises) = 0 and #(falls) = 2, so 208 is in the sequence.

Links

Crossrefs

Cf. A296762, A296763, A296712.

Programs

  • Mathematica
    z = 200; b = 20; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296762 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296763 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296764 *)

A296753 Numbers whose base-14 digits d(m), d(m-1), ..., d(0) have #(rises) = #(falls); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 211, 224, 225, 238, 239, 240, 252, 253, 254, 255, 266, 267, 268, 269, 270, 280, 281, 282
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2018

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296753-A296755 partition the natural numbers. See the guide at A296712.

Examples

			The base-14 digits of 1000000 are 2,8,6,2,12; here #(rises) = 2 and #(falls) = 2, so 1000000 is in the sequence.

Links

Crossrefs

Cf. A296754, A296755, A296712.

Programs

  • Mathematica
    z = 200; b = 14; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296753 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296754 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296755 *)
Select[Range[300],Total[Sign[Differences[IntegerDigits[#,14]]]]==0&] (* Harvey P. Dale, Sep 20 2022 *)

A296897 Numbers whose base-15 digits d(m), d(m-1), ..., d(0) have #(pits) = #(peaks); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296897-A296899 partition the natural numbers. See the guides at A296712 and A296882.

Examples

			The base-15 digits of 3842 are 1,2,1,2; here #(pits) = 1 and #(peaks) = 1, so 3842 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296898, A296899.

Programs

  • Mathematica
    z = 200; b = 15;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296897 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296898 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296899 *)

A296700 Numbers whose base-6 digits d(m), d(m-1), ... d(0) have #(rises) = #(falls); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 37, 38, 39, 40, 41, 43, 48, 49, 54, 55, 56, 60, 61, 62, 63, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 80, 81, 82, 83, 86, 90, 91, 92, 96, 97, 98, 99, 102, 103, 104, 105, 106, 109, 110, 111, 112, 113, 116, 117, 118, 119, 123
Offset: 1

Views

Author

Clark Kimberling, Dec 21 2017

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296700-A296702 partition the natural numbers. See the guide at A296712.

Examples

			The base-6 digits of 123 are 3,2,3; here #(rises) = 1 and #(falls) = 1, so 123 is in the sequence.

Links

Crossrefs

Cf. A296701, A296702, A296712.

Programs

  • Mathematica
    z = 200; b = 6; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296700 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296701 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296702 *)

A296709 Numbers whose base-9 digits d(m), d(m-1), ..., d(0) have #(rises) = #(falls); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 99, 100, 108, 109, 110, 117, 118, 119, 120, 126, 127, 128, 129, 130, 135, 136, 137, 138, 139, 140, 144, 145, 146, 147, 148, 149, 150, 153, 154, 155, 156, 157, 158
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2018

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296709-A296711 partition the natural numbers. See the guide at A296712.

Examples

			The base-9 digits of 158 are 1,8,5; here #(rises) = 1 and #(falls) = 1, so 158 is in the sequence.

Links

Crossrefs

Cf. A296710, A296711, A296712.

Programs

  • Mathematica
    z = 200; b = 9; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296709 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296710 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296711 *)

A296749 Numbers whose base-12 digits d(m), d(m-1), ..., d(0) have #(rises) < #(falls); see Comments.

Original entry on oeis.org

12, 24, 25, 36, 37, 38, 48, 49, 50, 51, 60, 61, 62, 63, 64, 72, 73, 74, 75, 76, 77, 84, 85, 86, 87, 88, 89, 90, 96, 97, 98, 99, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 116, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 132, 133, 134
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2018

Keywords

Comments

A rise is an index i such that d(i) < d(i+1); a fall is an index i such that d(i) > d(i+1). The sequences A296747-A296749 partition the natural numbers. See the guide at A296712.

Examples

			The base-12 digits of 134 are 11,2; here #(rises) = 0 and #(falls) = 2, so 134 is in the sequence.

Links

Crossrefs

Cf. A296747, A296748, A296712.

Programs

  • Mathematica
    z = 200; b = 12; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296747 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296748 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296749 *)
Showing 1-10 of 115 results. Next

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