A297241 - OEIS

%I #7 Jan 19 2018 16:22:54

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11,12,0,0,0,1,2,3,

%T 4,5,6,7,8,9,10,11,0,0,0,0,1,2,3,4,5,6,7,8,9,10,0,0,0,0,0,1,2,3,4,5,6,

%U 7,8,9,0,0,0,0,0,0,1,2,3,4,5,6,7,8,0

%N Up-variation of the base-14 digits of n; see Comments.

%C Suppose that a number 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). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.

%H Clark Kimberling, <a href="/A297241/b297241.txt">Table of n, a(n) for n = 1..10000</a>

%e 18 in base 14: 1,4; here UV = 3, so that a(18) = 3.

%t g[n_, b_] := Differences[IntegerDigits[n, b]];

%t b = 14; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297240 *)

%t Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297241 *)

%Y Cf. A297240, A297242, A297330.

%K nonn,base,easy

%O 1,17

%A _Clark Kimberling_, Jan 17 2018