A151950 -id:A151950 - 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

A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 18, 27, 36, 45, 54, 63, 72, 18, 9, 0, 9, 18, 27, 36, 45, 54, 63, 27, 18, 9, 0, 9, 18, 27, 36, 45, 54, 36, 27, 18, 9, 0, 9, 18, 27, 36, 45, 45, 36, 27, 18, 9, 0, 9, 18, 27, 36, 54, 45, 36, 27, 18, 9, 0, 9, 18, 27, 63, 54, 45, 36, 27, 18, 9, 0, 9, 18, 72, 63, 54, 45, 36, 27, 18, 9, 0, 9, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 99, 99, 198, 297, 396, 495, 594, 693, 792, 891, 99, 0, 99, 198, 297, 396, 495, 594, 693, 792

Offset: 0

N. J. A. Sloane, Aug 18 2009

Entries are multiples of 9 - see A151950.

a(n) = A004186(n) - A004185(n); a(A010785(n)) = 0. - Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

			For n = 15, a(15) = 51 - 15 = 36. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (terms 0..1000 from Joseph Myers and Robert G. Wilson v)
H. Hanslik, E. Hetmaniok, I. Sobstyl, et al., Orbits of the Kaprekar's transformations-some introductory facts, Zeszyty Naukowe Politechniki Śląskiej, Seria: Matematyka Stosowana z. 5, Nr kol. 1945; 2015.
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.
R. J. Mathar, Maple code for A151949 and A151959
Joseph Myers, C program for computing sequences related to the Kaprekar map
Joseph Myers, List of cycles under Kaprekar map (all numbers with <= 60 digits; cycles are represented by their smallest value)
Index entries for the Kaprekar map

In other bases: A164884 (base 2), A164993 (base 3), A165012 (base 4), A165032 (base 5), A165051 (base 6), A165071 (base 7), A165090 (base 8), A165110 (base 9). - Joseph Myers, Sep 05 2009
Cf. A151959, A069746, A347688.
Cf. also A004185, A004186, A099009 (fixed points).

a151949 n = a004186 n - a004185 n
-- Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

f[n_] := Module[{idn = IntegerDigits@n, idns}, idns = Sort@ idn; FromDigits@ Reverse@ idns - FromDigits@ idns]; Table[ f@n, {n, 0, 200}] (* Harvey P. Dale, Aug 18 2009 *)
Flatten[Table[Differences[FromDigits /@ {y = Sort[x = IntegerDigits[n]], Reverse[y]}], {n, 0, 74}]] (* Jayanta Basu, Jul 11 2013 *)

a(n) = {my(d=digits(n)); fromdigits(vecsort(d,,4)) - fromdigits(vecsort(d));} \\ Michel Marcus, Dec 08 2019

def A151949(n):
    return int("".join(sorted(str(n),reverse=True)))-int("".join(sorted(str(n)))) # Indranil Ghosh, Feb 01 2017

More terms from Robert G. Wilson v, Aug 19 2009

More than the usual number of terms are shown in order to distinguish this from similar sequences. - N. J. A. Sloane, Sep 22 2021

A165033 A165032(n)/4.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 4, 3, 2, 1, 0, 6, 6, 12, 18, 24, 6, 0, 6, 12, 18, 12, 6, 6, 12, 18, 18, 12, 12, 12, 18, 24, 18, 18, 18, 18, 12, 12, 12, 18, 24, 12, 6, 6, 12, 18, 12, 6, 0, 6, 12, 18, 12, 6, 6, 12, 24, 18, 12, 12, 12, 18, 18, 18, 18, 24, 18, 12, 12

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..3125
Index entries for the Kaprekar map

Cf. A165032.

In other bases: A164884 (base 2), A164994 (base 3), A165013 (base 4), A165052 (base 6), A165072 (base 7), A165091 (base 8), A165111 (base 9), A151950 (base 10).

A164994 A164993(n)/2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 0, 4, 4, 8, 4, 0, 4, 8, 4, 4, 8, 8, 8, 8, 4, 4, 8, 4, 0, 13, 16, 29, 16, 13, 26, 29, 26, 29, 16, 13, 26, 13, 0, 13, 26, 13, 16, 29, 26, 29, 26, 13, 16, 29, 16, 13, 26, 29, 32, 29, 26, 29, 32, 29, 26, 29, 26, 29, 26, 13, 16, 29, 16, 13, 32, 29, 26, 29, 16, 13, 26

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2187
Index entries for the Kaprekar map

Cf. A164993.

In other bases: A164884 (base 2), A165013 (base 4), A165033 (base 5), A165052 (base 6), A165072 (base 7), A165091 (base 8), A165111 (base 9), A151950 (base 10).

A165013 a(n) = A165012(n)/3.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 0, 1, 3, 2, 1, 0, 5, 5, 10, 15, 5, 0, 5, 10, 10, 5, 5, 10, 15, 10, 10, 10, 10, 10, 10, 15, 10, 5, 5, 10, 10, 5, 0, 5, 15, 10, 5, 5, 15, 15, 15, 15, 15, 10, 10, 10, 15, 10, 5, 5, 15, 10, 5, 0, 21, 25, 46, 67, 25, 21, 42, 63, 46, 42, 46, 67, 67, 63, 67, 71, 25, 21

Offset: 0

Joseph Myers, Sep 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..8192 (terms 0..1024 from Joseph Myers)
Index entries for the Kaprekar map

Cf. A165012.

In other bases: A164884 (base 2), A164994 (base 3), A165033 (base 5), A165052 (base 6), A165072 (base 7), A165091 (base 8), A165111 (base 9), A151950 (base 10).

A165052 A165051(n)/5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 4, 3, 2, 1, 0, 1, 5, 4, 3, 2, 1, 0, 7, 7, 14, 21, 28, 35, 7, 0, 7, 14, 21, 28, 14, 7, 7, 14, 21, 28, 21, 14, 14, 14, 21, 28, 28, 21, 21, 21, 21, 28, 35, 28, 28, 28, 28, 28, 14, 14, 14, 21, 28, 35, 14, 7, 7, 14, 21, 28, 14

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..1296
Index entries for the Kaprekar map

Cf. A165051.

In other bases: A164884 (base 2), A164994 (base 3), A165013 (base 4), A165033 (base 5), A165072 (base 7), A165091 (base 8), A165111 (base 9), A151950 (base 10).

A165072 A165071(n)/6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 5, 4, 3, 2, 1, 0, 1, 6, 5, 4, 3, 2, 1, 0, 8, 8, 16, 24, 32, 40, 48, 8, 0, 8, 16, 24, 32, 40, 16, 8, 8, 16, 24, 32, 40, 24, 16, 16, 16, 24, 32, 40, 32, 24, 24, 24, 24, 32, 40, 40, 32, 32, 32, 32

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2401
Index entries for the Kaprekar map

Cf. A165071.

In other bases: A164884 (base 2), A164994 (base 3), A165013 (base 4), A165033 (base 5), A165052 (base 6), A165091 (base 8), A165111 (base 9), A151950 (base 10).

A165091 A165090(n)/7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 2, 1, 0, 1, 2, 3, 4, 5, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 5, 4, 3, 2, 1, 0, 1, 2, 6, 5, 4, 3, 2, 1, 0, 1, 7, 6, 5, 4, 3, 2, 1, 0, 9, 9, 18, 27, 36, 45, 54, 63, 9, 0, 9, 18, 27, 36, 45, 54, 18, 9, 9, 18, 27, 36, 45, 54, 27, 18, 18, 18, 27, 36

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..4096
Index entries for the Kaprekar map

Cf. A165090.

In other bases: A164884 (base 2), A164994 (base 3), A165013 (base 4), A165033 (base 5), A165052 (base 6), A165072 (base 7), A165111 (base 9), A151950 (base 10).

A165111 A165110(n)/8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 2, 1, 0, 1, 2, 3, 4, 5, 6, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 6, 5, 4, 3, 2, 1, 0, 1, 2, 7, 6, 5, 4, 3, 2, 1, 0, 1, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 10, 20, 30, 40, 50, 60, 70, 80, 10, 0, 10, 20, 30, 40, 50

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..6561
Index entries for the Kaprekar map

Cf. A165110.

In other bases: A164884 (base 2), A164994 (base 3), A165013 (base 4), A165033 (base 5), A165052 (base 6), A165072 (base 7), A165091 (base 8), A151950 (base 10).

A151951 a(1) = 1113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

Original entry on oeis.org

1113, 1998, 8082, 8532, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174

Offset: 1

N. J. A. Sloane, Aug 18 2009

Index entries for the Kaprekar map

Cf. A099009, A151947, A151949, A151950, A069746.

NestList[FromDigits[Reverse[Sort[IntegerDigits[#]]]]-FromDigits[Sort[ IntegerDigits[ #]]]&,1113,40] (* or *) PadRight[{1113,1998,8082,8532},40,{6174}] (* Harvey P. Dale, May 10 2021 *)

cp's OEIS Frontend

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

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A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

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A165033 A165032(n)/4.

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A164994 A164993(n)/2.

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A165013 a(n) = A165012(n)/3.

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A165052 A165051(n)/5.

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A165072 A165071(n)/6.

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A165091 A165090(n)/7.

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A165111 A165110(n)/8.

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A151951 a(1) = 1113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

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