A297271 Numbers whose base-10 digits have equal down-variation and up-variation; see Comments.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484

Offset: 1

Author

Clark Kimberling, Jan 16 2018

Keywords

• nonn
• base
• easy

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 after the zero from A002113 first at 1011, which is not a palindrome but has DV(1011,10) = UV(1011,10) =1. - R. J. Mathar, Jan 23 2018

Apart from 0, the initial terms coincide with those of A266140, but the two sequences are different. First disagreement: a(109) = 1001 and A266140(110) = 1111. - Georg Fischer, Oct 09 2018

Examples

			13601 in base-10:  1,3,6,0,1, having DV = 6, UV = 6, so that 13601 is in the sequence.
		

Links

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

Crossrefs

Cf. A002113, A266140, A297330, A297270, A297272.

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 = 10; 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]   (* A297270 *)
Take[Flatten[Position[w, 0]], 120]    (* A297271 *)
Take[Flatten[Position[w, 1]], 120]    (* A297272 *)

Formula

{k: A037851(k) = A037860(k)}. - R. J. Mathar, Sep 27 2021