A296712 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(rises) = #(falls); see Comments.

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

Author

Clark Kimberling, Jan 08 2018

Keywords

• nonn
• base
• easy

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.

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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

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

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 *)