A297281 - OEIS

A297281

Numbers whose base-13 digits have greater up-variation than down-variation; see Comments.

15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 43, 44, 45, 46, 47, 48, 49, 50, 51, 57, 58, 59, 60, 61, 62, 63, 64, 71, 72, 73, 74, 75, 76, 77, 85, 86, 87, 88, 89, 90, 99, 100, 101, 102, 103, 113, 114, 115, 116, 127, 128

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OFFSET

1,1

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 from A296751 for example at 171 = 102_13, which is in this sequence because UV(171,13) = 2 > DV(171,13)=1, but not in A296751 because the number of rises and falls are equal. - R. J. Mathar, Jan 23 2018

LINKS

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

EXAMPLE

128 in base-13: 9,11, having DV = 0, UV = 2, so that 28 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 *)