A296712 -id:A296712 - cp's OEIS frontend

A296902 Numbers whose base-16 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

288, 289, 304, 305, 306, 320, 321, 322, 323, 336, 337, 338, 339, 340, 352, 353, 354, 355, 356, 357, 368, 369, 370, 371, 372, 373, 374, 384, 385, 386, 387, 388, 389, 390, 391, 400, 401, 402, 403, 404, 405, 406, 407, 408, 416, 417, 418, 419, 420, 421, 422, 423

Offset: 1

Clark Kimberling, Jan 12 2018

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 A296900-A296902 partition the natural numbers. See the guides at A296712 and A296882.

			The base-16 digits of 74017 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so 74017 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296900, A296901.

z = 200; b = 16;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296900 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296901 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296902 *)

A296904 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

Original entry on oeis.org

401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835

Offset: 1

Clark Kimberling, Jan 12 2018

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 A296903..A296905 partition the natural numbers. See the guides at A296712 and A296882.

			The base-20 digits of 328822 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so 328822 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296904, A296905.

z = 200; b = 20;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296903 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296904 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296905 *)

A296905 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

Original entry on oeis.org

440, 441, 460, 461, 462, 480, 481, 482, 483, 500, 501, 502, 503, 504, 520, 521, 522, 523, 524, 525, 540, 541, 542, 543, 544, 545, 546, 560, 561, 562, 563, 564, 565, 566, 567, 580, 581, 582, 583, 584, 585, 586, 587, 588, 600, 601, 602, 603, 604, 605, 606, 607

Offset: 1

Clark Kimberling, Jan 12 2018

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 A296903..A296905 partition the natural numbers. See the guides at A296712 and A296882.

			The base-20 digits of 176441 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so 176441 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296903, A296904.

z = 200; b = 20;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296903 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296904 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296905 *)

b-file replaced by Clark Kimberling, Feb 27 2018

A296907 Numbers whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

Original entry on oeis.org

3601, 3602, 3603, 3604, 3605, 3606, 3607, 3608, 3609, 3610, 3611, 3612, 3613, 3614, 3615, 3616, 3617, 3618, 3619, 3620, 3621, 3622, 3623, 3624, 3625, 3626, 3627, 3628, 3629, 3630, 3631, 3632, 3633, 3634, 3635, 3636, 3637, 3638, 3639, 3640, 3641, 3642, 3643

Offset: 1

Clark Kimberling, Jan 12 2018

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 A296906..A296908 partition the natural numbers. See the guides at A296712 and A296882.

			The base-60 digits of 26143262 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so 26143262 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296906, A296908.

z = 200; b = 60;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296906 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296907 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296908 *)

A296908 Numbers whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

Original entry on oeis.org

3720, 3721, 3780, 3781, 3782, 3840, 3841, 3842, 3843, 3900, 3901, 3902, 3903, 3904, 3960, 3961, 3962, 3963, 3964, 3965, 4020, 4021, 4022, 4023, 4024, 4025, 4026, 4080, 4081, 4082, 4083, 4084, 4085, 4086, 4087, 4140, 4141, 4142, 4143, 4144, 4145, 4146, 4147

Offset: 1

Clark Kimberling, Jan 12 2018

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 A296906..A296908 partition the natural numbers. See the guides at A296712 and A296882.

			The base-60 digits of 13395721 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so 13395721 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296906, A296907.

z = 200; b = 60;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296906 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296907 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296908 *)

A296713 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(rises) > #(falls); see Comments.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129, 133, 134, 135, 136, 137, 138, 139

Offset: 1

Clark Kimberling, Jan 08 2018

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 A296712-A296714 partition the natural numbers. See the guide at A296712.

			The base-10 digits of 139 are 1,3,9; here #(rises) = 2 and #(falls) = 0, so 139 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296712, A296714, A296712.

z = 200; b = 10; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296712 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296713 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296714 *)
Select[Range[150],Total[Sign[Differences[IntegerDigits[#]]]]>0&] (* Harvey P. Dale, May 21 2021 *)

A296714 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(rises) < #(falls); see Comments.

Original entry on oeis.org

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 200, 210, 211, 220, 221, 300, 310, 311, 320, 321, 322, 330, 331, 332

Offset: 1

Clark Kimberling, Jan 08 2018

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 A296712-A296714 partition the natural numbers. See the guide at A296712.

			The base-10 digits of 332 are 3,3,2; here #(rises) = 0 and #(falls) = 1, so 332 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296712, A296713, A296712.

z = 200; b = 10; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296712 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296713 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296714 *)

A296883 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

Original entry on oeis.org

101, 102, 103, 104, 105, 106, 107, 108, 109, 201, 202, 203, 204, 205, 206, 207, 208, 209, 212, 213, 214, 215, 216, 217, 218, 219, 301, 302, 303, 304, 305, 306, 307, 308, 309, 312, 313, 314, 315, 316, 317, 318, 319, 323, 324, 325, 326, 327, 328, 329, 401, 402

Offset: 1

Clark Kimberling, Jan 10 2018

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 and A296882.

			The base-10 digits of 21212 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so 21212 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296882, A296884.

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 *)

A296884 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

Original entry on oeis.org

120, 121, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195, 196, 197, 198, 230, 231, 232, 240, 241, 242, 243, 250

Offset: 1

Clark Kimberling, Jan 10 2018

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 and A296882.

			The base-10 digits of 12121 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so 12121 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A296882, A296712, A296882, A296883.

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 *)

A297031 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-3 digits of n; see Comments.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 1, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 2, 1, 2, 3, 3, 4, 4, 3

Offset: 1

Clark Kimberling, Jan 13 2018

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. See A297030 for a guide to related sequences.

			Base-3 digits for 123:  1, 1, 1, 2, 0, so that a(123) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297030 (pieces), A296712 (rises and falls), A296882 (pits and peaks).

a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
b = 3; Table[a[n, b], {n, 1, 120}]

cp's OEIS Frontend

A296902 Numbers whose base-16 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

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A296904 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

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A296905 Numbers whose base-20 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

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A296907 Numbers whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

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A296908 Numbers whose base-60 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

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A296713 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(rises) > #(falls); see Comments.

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A296714 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(rises) < #(falls); see Comments.

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A296883 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

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