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A297289 Numbers whose base-16 digits have equal down-variation and up-variation; see Comments.

%I A297289 #8 Jan 23 2018 20:40:24
%S A297289 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,34,51,68,85,102,119,136,153,
%T A297289 170,187,204,221,238,255,257,273,289,305,321,337,353,369,385,401,417,
%U A297289 433,449,465,481,497,514,530,546,562,578,594,610,626,642,658,674
%N A297289 Numbers whose base-16 digits have equal down-variation and up-variation; see Comments.
%C A297289 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 A297289 Differs from A029730 after the zero first at 4113 = 1011_16 (not a base-16 palindrome), where DV=UV=1. - _R. J. Mathar_, Jan 23 2018
%H A297289 Clark Kimberling, <a href="/A297289/b297289.txt">Table of n, a(n) for n = 1..10000</a>
%e A297289 674 in base-16:  2,10,2 having DV = 8, UV = 8, so that 674 is in the sequence.
%t A297289 g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t A297289 x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t A297289 b = 16; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t A297289 w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t A297289 Take[Flatten[Position[w, -1]], 120]   (* A297288 *)
%t A297289 Take[Flatten[Position[w, 0]], 120]    (* A297289 *)
%t A297289 Take[Flatten[Position[w, 1]], 120]    (* A297290 *)
%Y A297289 Cf. A297330, A297288, A297290.
%K A297289 nonn,base,easy
%O A297289 1,2
%A A297289 _Clark Kimberling_, Jan 17 2018