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Numbers whose base-10 digits have greater down-variation than up-variation; see Comments.
4

%I #9 Jan 23 2018 19:35:11

%S 10,20,21,30,31,32,40,41,42,43,50,51,52,53,54,60,61,62,63,64,65,70,71,

%T 72,73,74,75,76,80,81,82,83,84,85,86,87,90,91,92,93,94,95,96,97,98,

%U 100,110,120,130,140,150,160,170,180,190,200,201,210,211,220,221

%N Numbers whose base-10 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.

%C Differs first from A071590 at 1101, which is in A071590, but not in here because UV(1101) = DV(1101). - _R. J. Mathar_, Jan 23 2018

%H Clark Kimberling, <a href="/A297270/b297270.txt">Table of n, a(n) for n = 1..10000</a>

%e 6151413121 in base-10: 6,1,5,1,4,1,3,1,2,1, having DV = 15, UV = 10, so that 6151413121 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 = 10; 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] (* A297270 *)

%t Take[Flatten[Position[w, 0]], 120] (* A297271 *)

%t Take[Flatten[Position[w, 1]], 120] (* A297272 *)

%Y Cf. A297330, A297271, A297272.

%K nonn,base,easy

%O 1,1

%A _Clark Kimberling_, Jan 16 2018