A037856 -id:A037856 - cp's OEIS frontend

A297330 Total variation of base-10 digits of n; see Comments.

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

Offset: 1

Clark Kimberling, Jan 17 2018

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)}
         A033264     A037800      A037834
         A037853     A037844      A037835
         A037854     A037845      A037836
         A037855     A037846      A037837
         A037856     A037847      A037838
         A037857     A037848      A037839
         A037858     A037849      A037840
         A037859     A037850      A037841
        A037860     A037851      A297330
        A297231     A297232      A297233
        A297234     A297235      A297236
        A297237     A297238      A297239
        A297240     A297241      A297242
        A297243     A297244      A297245
        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
         A005843      A005408      (none)
         A297249      A297250      A297251
         A297252      A297253      A297254
         A297255      A297256      A297257
         A297258      A297259      A297260
         A297261      A297262      A297263
         A297264      A297265      A297266
         A297267      A297268      A297269
        A297270      A297271      A297272
        A297273      A297274      A297275
        A297276      A297277      A297278
        A297279      A297280      A297281
        A297282      A297283      A297284
        A297285      A297286      A297287
        A297288      A297289      A297290
Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - Robert Israel, Feb 06 2018

			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.

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

Cf. A037851, A297330, A297271, A297272.

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
# alternative
A297330 := proc(n)
    A037860(n)+A037851(n) ;
end proc: # R. J. Mathar, Sep 27 2021

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 *)

def A297330(n):
    s = str(n)
    return sum(abs(int(s[i])-int(s[i+1])) for i in range(len(s)-1)) # Chai Wah Wu, May 31 2022

A037847 a(n)=Sum{d(i-1)-d(i): d(i)

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 1, 2, 3, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 2, 3, 4, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 0, 0, 0, 1, 2, 3, 1

Offset: 1

Clark Kimberling

This is the base-6 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

A037847 := proc(n)
    a := 0 ;
    dgs := convert(n,base,6);
    for i from 2 to nops(dgs) do
        if op(i,dgs)R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 6; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037856*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037847*)

Definition swapped with A037856. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A037874 (1/2)Sum{|d(i)-e(i)|}, where Sum{d(i)6^i} is base 6 representation of n and e(i) are digits d(i) in nonincreasing order, for i=0,1,...,m.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 4, 3, 2, 1, 0, 0, 5, 4, 3, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 4, 3, 2, 1, 0, 0, 5, 4, 3, 2, 1, 0, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 3

Offset: 1

Clark Kimberling

Starts to differ from A037856 at n=220 = 1010_6. - R. J. Mathar, Aug 05 2025

A037874 := proc(n)
    local a,dgs,dgsE ;
    a := 0 ;
    dgs := convert(n,base,6);
    dgsE := sort(dgs,`>`) ;
    for i from 1 to nops(dgs) do
        a := a+ abs(op(i,dgs)-op(i,dgsE)) ;
    end do:
    a/2 ;
end proc: # R. J. Mathar, Oct 19 2015

Definition swapped with A037883. - R. J. Mathar, Oct 19 2015

cp's OEIS Frontend

A297330 Total variation of base-10 digits of n; see Comments.

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A037847 a(n)=Sum{d(i-1)-d(i): d(i)

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A037874 (1/2)Sum{|d(i)-e(i)|}, where Sum{d(i)6^i} is base 6 representation of n and e(i) are digits d(i) in nonincreasing order, for i=0,1,...,m.

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cp's OEIS Frontend

A297330 Total variation of base-10 digits of n; see Comments.

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Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

Python

A037847 a(n)=Sum{d(i-1)-d(i): d(i)

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Maple

Mathematica

Extensions

A037874 (1/2)*Sum{|d(i)-e(i)|}, where Sum{d(i)*6^i} is base 6 representation of n and e(i) are digits d(i) in nonincreasing order, for i=0,1,...,m.

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Maple

Extensions

A037874 (1/2)Sum{|d(i)-e(i)|}, where Sum{d(i)6^i} is base 6 representation of n and e(i) are digits d(i) in nonincreasing order, for i=0,1,...,m.