A297277 Numbers whose base-12 digits have equal down-variation and up-variation; see Comments.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555

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 first from A029957 after the zero for 1741 = 1011_12, which is not a palindrome in base 12 but has DV(1741,12) = UV(1741,12) =1. - R. J. Mathar, Jan 23 2018

Examples

			555 in base-12:  3,10,3, having DV = 7, UV = 7, so that 555 is in the sequence.
		

Links

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

Crossrefs

Cf. A297330, A297276, A297278.

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 = 12; 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]   (* A297276 *)
Take[Flatten[Position[w, 0]], 120]    (* A297277 *)
Take[Flatten[Position[w, 1]], 120]    (* A297278 *)