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

1,2

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

Links

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

Example

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

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 *)

Crossrefs

Cf. A297330, A297276, A297278.

Sequence in context: A043715 A296747 A029957 * A048323 A048336 A130574

Adjacent sequences: A297274 A297275 A297276 * A297278 A297279 A297280

Keyword

nonn,base,easy

Author

Clark Kimberling, Jan 16 2018

Status

approved