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A297265 Numbers whose base-8 digits have equal down-variation and up-variation; see Comments.

%I A297265 #8 Nov 04 2018 01:44:47
%S A297265 1,2,3,4,5,6,7,9,18,27,36,45,54,63,65,73,81,89,97,105,113,121,130,138,
%T A297265 146,154,162,170,178,186,195,203,211,219,227,235,243,251,260,268,276,
%U A297265 284,292,300,308,316,325,333,341,349,357,365,373,381,390,398,406
%N A297265 Numbers whose base-8 digits have equal down-variation and up-variation; see Comments.
%C A297265 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 A297265 a(n) = A029803(n+1) for 1 <= n < 72, but a(72) = 521 differs from A029803(73) = 585. - _Georg Fischer_, Oct 30 2018
%H A297265 Clark Kimberling, <a href="/A297265/b297265.txt">Table of n, a(n) for n = 1..10000</a>
%e A297265 406 in base-8:  6,2,6, having DV = 4, UV = 4, so that 406 is in the sequence.
%t A297265 g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t A297265 x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t A297265 b = 8; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t A297265 w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t A297265 Take[Flatten[Position[w, -1]], 120]   (* A297264 *)
%t A297265 Take[Flatten[Position[w, 0]], 120]    (* A297265 *)
%t A297265 Take[Flatten[Position[w, 1]], 120]    (* A297266 *)
%Y A297265 Cf. A297330, A297264, A297266.
%K A297265 nonn,base,easy
%O A297265 1,2
%A A297265 _Clark Kimberling_, Jan 15 2018