A297280 - OEIS

A297280

Numbers whose base-13 digits have equal down-variation and up-variation; see Comments.

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, 183, 196, 209, 222, 235, 248, 261, 274, 287, 300, 313, 326, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 510, 523, 536, 549, 562, 575, 588

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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 after the zero from A029958 first for 2211 = 1011_13, which is not a palindrome in base 13 but has DV(2211,13) = UV(2211,13) =1. - R. J. Mathar, Jan 23 2018

LINKS

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

EXAMPLE

588 in base-13: 3,6,3, having DV = 3, UV = 3, so that 588 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 = 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 *)