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Numbers whose base-10 digits have greater up-variation than down-variation; see Comments.
4

%I #8 Jan 23 2018 19:41:32

%S 12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,34,35,36,37,38,39,45,46,

%T 47,48,49,56,57,58,59,67,68,69,78,79,89,102,103,104,105,106,107,108,

%U 109,112,113,114,115,116,117,118,119,122,123,124,125,126,127,128

%N Numbers whose base-10 digits have greater up-variation than down-variation; see Comments.

%C 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.

%C Differs from A071589 first at 1011 which is in A071589 but not in here because UV(1011) = DV(1011)=1. - _R. J. Mathar_, Jan 23 2018

%H Clark Kimberling, <a href="/A297272/b297272.txt">Table of n, a(n) for n = 1..10000</a>

%e 198765 in base-10: 1,9,8,7,6,5, having DV = 4, UV = 8, so that 198765 is in the sequence.

%t g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];

%t x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];

%t b = 10; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];

%t w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];

%t Take[Flatten[Position[w, -1]], 120] (* A297270 *)

%t Take[Flatten[Position[w, 0]], 120] (* A297271 *)

%t Take[Flatten[Position[w, 1]], 120] (* A297272 *)

%Y Cf. A297330, A297270, A297271.

%K nonn,base,easy

%O 1,1

%A _Clark Kimberling_, Jan 16 2018