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A297330 Total variation of base-10 digits of n; see Comments. 89
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

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).  Guide to related sequences and partitions of the natural numbers:

***

Base b     {DV(n,b)}   {UV(n,b)}    {TV(n,b)}

2           A033264     A037800      A037834

3           A037853     A037844      A037835

4           A037854     A037845      A037836

5           A037855     A037846      A037837

6           A037856     A037847      A037838

7           A037857     A037848      A037839

8           A037858     A037849      A037840

9           A037859     A037850      A037841

10          A037860     A037851      A297330

11          A297231     A297232      A297233

12          A297234     A297235      A297236

13          A297237     A297238      A297239

14          A297240     A297241      A297242

15          A297243     A297244      A297245

16          A297246     A297247      A297247

For each b, let u = {n : UV(n,b) < DV(n,b)}, e = {n : UV(n,b) = DV(n,b)}, and d = {n : UV(n,b) > DV(n,b)}.  The sets u,e,d partition the natural numbers.  A guide to the matching sequences for u, e, d follows:

***

Base b    Sequence u   Sequence e   Sequence d

2           A005843      A005408      (none)

3           A297249      A297250      A297251

4           A297252      A297253      A297254

5           A297255      A297256      A297257

6           A297258      A297259      A297260

7           A297261      A297262      A297263

8           A297264      A297265      A297266

9           A297267      A297268      A297269

10          A297270      A297271      A297272

11          A297273      A297274      A297275

12          A297276      A297277      A297278

13          A297279      A297280      A297281

14          A297282      A297283      A297284

15          A297285      A297286      A297287

16          A297288      A297289      A297290

Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - Robert Israel, Feb 06 2018

LINKS

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

EXAMPLE

13684632 has DV = 8-4 + 6-3 + 3-2 = 8 and has UV = 3-1 + 6-3 + 8-6 + 6-4 = 9, so that a(13684632) = DV + UV = 17.

MAPLE

f:= proc(n) local L, i; L:= convert(n, base, 10);

add(abs(L[i+1]-L[i]), i=1..nops(L)-1) end proc:

map(f, [$1..100]); # Robert Israel, Feb 04 2018

MATHEMATICA

b = 10; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[ IntegerDigits[n, b], 2, 1]], {n, z}] (* after Michael De Vlieger, e.g. A037834 *)

CROSSREFS

Cf. A037851, A297330, A297271, A297272.

Sequence in context: A255594 A030108 A307651 * A037904 A070615 A040114

Adjacent sequences:  A297327 A297328 A297329 * A297331 A297332 A297333

KEYWORD

nonn,base,easy

AUTHOR

Clark Kimberling, Jan 17 2018

STATUS

approved

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Last modified June 17 15:24 EDT 2019. Contains 324189 sequences. (Running on oeis4.)