A297284 Numbers whose base-14 digits have greater up-variation than down-variation; see Comments.

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 67, 68, 69, 76, 77, 78, 79, 80, 81, 82, 83, 91, 92, 93, 94, 95, 96, 97, 106, 107, 108, 109, 110, 111

Offset: 1

Author

Clark Kimberling, Jan 17 2018

Keywords

• nonn
• base
• easy

Comments

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.

Differs from A296754 first at 198 =102_14, which is in this sequence because UV(102,14) = 2 > DV(102,14) =1, but has the same number of rises and falls and is not in A296754. - R. J. Mathar, Jan 23 2018

Examples

			111 in base-14:  7,13 having DV = 0, UV = 6, so that 111 is in the sequence.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A297330, A297282, A297283.

Programs

Mathematica

g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 14; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297282 *)
Take[Flatten[Position[w, 0]], 120]    (* A297283 *)
Take[Flatten[Position[w, 1]], 120]    (* A297284 *)