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A297263 Numbers whose base-7 digits have greater up-variation than down-variation; see Comments.

%I A297263 #4 Jan 16 2018 11:31:31
%S A297263 9,10,11,12,13,17,18,19,20,25,26,27,33,34,41,51,52,53,54,55,58,59,60,
%T A297263 61,62,65,66,67,68,69,72,73,74,75,76,79,80,81,82,83,86,87,88,89,90,93,
%U A297263 94,95,96,97,101,102,103,104,108,109,110,111,115,116,117,118
%N A297263 Numbers whose base-7 digits have greater up-variation than down-variation; see Comments.
%C A297263 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.
%H A297263 Clark Kimberling, <a href="/A297263/b297263.txt">Table of n, a(n) for n = 1..10000</a>
%e A297263 118 in base-7:  2,2,6, having DV = 0, UV = 4, so that 118 is in the sequence.
%t A297263 g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t A297263 x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t A297263 b = 7; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t A297263 w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t A297263 Take[Flatten[Position[w, -1]], 120]   (* A297261 *)
%t A297263 Take[Flatten[Position[w, 0]], 120]    (* A297262 *)
%t A297263 Take[Flatten[Position[w, 1]], 120]    (* A297263 *)
%Y A297263 Cf. A297330, A297261, A297262.
%K A297263 nonn,base,easy
%O A297263 1,1
%A A297263 _Clark Kimberling_, Jan 15 2018