A297267 - OEIS

%I #4 Jan 16 2018 11:32:03

%S 9,18,19,27,28,29,36,37,38,39,45,46,47,48,49,54,55,56,57,58,59,63,64,

%T 65,66,67,68,69,72,73,74,75,76,77,78,79,81,90,99,108,117,126,135,144,

%U 153,162,163,171,172,180,181,189,190,198,199,207,208,216,217,225

%N Numbers whose base-9 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="/A297267/b297267.txt">Table of n, a(n) for n = 1..10000</a>

%e 225 in base-9:  2,7,0, having DV = 7, UV = 5, so that 225 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 = 9; 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]   (* A297267 *)

%t Take[Flatten[Position[w, 0]], 120]    (* A297268 *)

%t Take[Flatten[Position[w, 1]], 120]    (* A297269 *)

%Y Cf. A297330, A297268, A297269.

%K nonn,base,easy

%O 1,1

%A _Clark Kimberling_, Jan 15 2018