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 1-4 of 4 results.

A296712 Numbers whose base-10 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, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 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
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 A296712-A296714 partition the natural numbers.
****
Guide to related sequences:
Base #(rises) = #(falls) #(rises) > #(falls) #(rises) < #(falls)
2 A005408 (none) A005843
3 A296691 A296692 A296693
4 A296694 A296695 A296696
5 A296697 A296698 A296699
6 A296700 A296701 A296702
7 A296703 A296704 A296705
8 A296706 A296707 A296708
9 A296709 A296710 A296711
10 A296712 A296713 A296714
11 A296744 A296745 A296746
12 A296747 A296748 A296749
13 A296750 A296751 A296752
14 A296753 A296754 A296755
15 A296756 A296757 A296758
16 A296759 A296760 A296761
20 A296762 A296763 A296764
60 A296765 A296766 A296767

Examples

			The base-10 digits of 181 are 1,8,1; here #(rises) = 1 and #(falls) = 1, so 181 is in the sequence.

Links

Crossrefs

Cf. A296713, A296714, A296712.

Programs

  • Mathematica
    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 *)

A296751 Numbers whose base-13 digits d(m), d(m-1), ..., d(0) have #(rises) > #(falls); see Comments.

Original entry on oeis.org

15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 43, 44, 45, 46, 47, 48, 49, 50, 51, 57, 58, 59, 60, 61, 62, 63, 64, 71, 72, 73, 74, 75, 76, 77, 85, 86, 87, 88, 89, 90, 99, 100, 101, 102, 103, 113, 114, 115, 116, 127, 128
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 A296751-A296753 partition the natural numbers. See the guide at A296712.

Examples

			The base-13 digits of 1000 are 5,11,12; here #(rises) = 2 and #(falls) = 0, so 1000 is in the sequence.

Links

Crossrefs

Cf. A296750, A296752, A296712.

Programs

  • Mathematica
    z = 200; b = 13; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296750 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296751 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296752 *)

A296750 Numbers whose base-13 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, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 183, 195, 196, 208, 209, 210, 221, 222, 223, 224, 234, 235, 236, 237, 238, 247, 248, 249, 250, 251, 252
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 A296750-A296751 partition the natural numbers. See the guide at A296712.

Examples

			The base-13 digits of 998 are 5,11,10; here #(rises) = 1 and #(falls) = 1, so 998 is in the sequence.

Links

Crossrefs

Cf. A296751, A296752, A296712.

Programs

  • Mathematica
    z = 200; b = 13; d[n_] := Sign[Differences[IntegerDigits[n, b]]];
Select[Range [z], Count[d[#], -1] == Count[d[#], 1] &] (* A296750 *)
Select[Range [z], Count[d[#], -1] < Count[d[#], 1] &]  (* A296751 *)
Select[Range [z], Count[d[#], -1] > Count[d[#], 1] &]  (* A296752 *)

A297279 Numbers whose base-13 digits have greater down-variation than up-variation; see Comments.

Original entry on oeis.org

13, 26, 27, 39, 40, 41, 52, 53, 54, 55, 65, 66, 67, 68, 69, 78, 79, 80, 81, 82, 83, 91, 92, 93, 94, 95, 96, 97, 104, 105, 106, 107, 108, 109, 110, 111, 117, 118, 119, 120, 121, 122, 123, 124, 125, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 143, 144
Offset: 1

Views

Author

Clark Kimberling, Jan 17 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.
Differs from A296752 first for 195 = 120_13, which has the same number of rises and falls and is therefore not in A296752, but has DV(195,13) =2 > UV(195,13) = 1 and is in this sequence. - R. J. Mathar, Jan 23 2018

Examples

			144 in base-13:  11,1, having DV = 10, UV = 0, so that 144 is in the sequence.

Links

Crossrefs

Cf. A297330, A297280, A297281.

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 = 13; 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]   (* A297279 *)
Take[Flatten[Position[w, 0]], 120]    (* A297280 *)
Take[Flatten[Position[w, 1]], 120]    (* A297281 *)
Showing 1-4 of 4 results.

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