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%I #4 Jan 15 2018 15:31:40
%S 4,8,9,12,13,14,16,20,24,28,32,33,36,37,40,41,44,45,48,49,50,52,53,54,
%T 56,57,58,60,61,62,64,68,72,76,80,84,88,92,96,100,104,108,112,116,120,
%U 124,128,129,132,133,136,137,140,141,144,145,148,149,152,153
%N Numbers whose base-4 digits have greater down-variation than up-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.
%H Clark Kimberling, <a href="/A297252/b297252.txt">Table of n, a(n) for n = 1..10000</a>
%e 153 in base-4: 2,1,2,1, having DV = 2, UV = 1, so that 153 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 = 4; 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] (* A297252 *)
%t Take[Flatten[Position[w, 0]], 120] (* A297253 *)
%t Take[Flatten[Position[w, 1]], 120] (* A297254 *)
%Y Cf. A297330, A297253, A297254.
%K nonn,base,easy
%O 1,1
%A _Clark Kimberling_, Jan 15 2018
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