A053844 -id:A053844 - cp's OEIS frontend

A053838 a(n) = (sum of digits of n written in base 3) modulo 3.

0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0

Offset: 0

Henry Bottomley, Mar 28 2000

Start with 0, repeatedly apply the morphism 0->012, 1->120, 2->201. This is a ternary version of the Thue-Morse sequence A010060. See Brlek (1989). - N. J. A. Sloane, Jul 10 2012

A090193 is generated by the same mapping starting with 1. A090239 is generated by the same mapping starting with 2. - Andrey Zabolotskiy, May 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
Arthur Dolgopolov, Equitable Sequencing and Allocation Under Uncertainty, Preprint, 2016.
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Michael Gilleland, Some Self-Similar Integer Sequences
Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016. See Example 17.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for sequences that are fixed points of mappings

Cf. A004128, A010060, A053837, A053839-A053844.

Equals A026600(n+1) - 1.

A053838 := proc(n)
    add(d,d=convert(n,base,3)) ;
    modp(%,3) ;
end proc:
seq(A053838(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Nest[ Flatten[ # /. {0 -> {0, 1, 2}, 1 -> {1, 2, 0}, 2 -> {2, 0, 1}}] &, {0}, 7] (* Robert G. Wilson v, Mar 08 2005 *)

a(n) = vecsum(digits(n, 3)) % 3; \\ Michel Marcus, May 04 2016

from sympy.ntheory import digits
def A053838(n): return sum(digits(n,3)[1:])%3 # Chai Wah Wu, Feb 28 2025

a(n) = A010872(A053735(n)) =(n+a(floor[n/3])) mod 3. So one can construct sequence by starting with 0 and mapping 0->012, 1->120 and 2->201 (e.g. 0, 012, 012120201, 012120201120201012201012120, ...) and looking at n-th digit of a term with sufficient digits.

a(n) = A004128(n) mod 3. [Gary W. Adamson, Aug 24 2008]

A053837 Sum of digits of n modulo 10.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

			a(59)=4 because 5+9 = 14 = 4 mod 10.

Cf. A007953, A010879.

Cf. A000120, A010060, A053838, A053839, A053840, A053841, A053842, A053843, A053844.

Table[Mod[Total[IntegerDigits[n]],10],{n,0,120}] (* Harvey P. Dale, Oct 02 2018 *)

a(n)=sumdigits(n)%10 \\ Charles R Greathouse IV, Feb 02 2023

a(n) = A010879(A007953(n)) = (n+a(floor[n/10])) mod 10. So can construct sequence by starting with 0 and mapping 0->0123456789, 1->1234567890, 2->2345678901 etc. (e.g. 0, 0123456789, 0123456789123456789023456789013456789012456..., etc.) and looking at n-th digit of a term with sufficient digits.

A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the third row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

This is the fixed point of the morphism 0->0123, 1->1230, 2->2301, 3->3012 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4, and t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. - Clark Kimberling, May 31 2017

			First three iterations of the morphism 0->0123, 1->1230, 2->2301, 3->3012:
  0123
  0123123023013012
  0123123023013012123023013012012323013012012312303012012312302301

Robert Israel, Table of n, a(n) for n = 0..10000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for sequences that are fixed points of mappings

Cf. A010060, A053837-A053844, A141803, A287552, A287553, A287554, A287555.

Cf. A010873, A053737.

seq(convert(convert(n,base,4),`+`) mod 4, n=0..100); # Robert Israel, May 18 2016

Mod[Total@ IntegerDigits[#, 4], 4] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)
s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];   (* - Clark Kimberling, May 31 2017 *)

a(n) = vecsum(digits(n,4)) % 4; \\ Michel Marcus, May 16 2016

a(n) = sumdigits(n, 4) % 4; \\ Michel Marcus, Jul 04 2018

a(n) = A010873(A053737(n)). - Andrey Zabolotskiy, May 18 2016

G.f. G(x) satisfies x^81*G(x) - (x^72+x^75+x^78+x^81)*G(x^4) + (x^48+x^60+x^63-x^64+x^72+x^75-x^76+x^78-x^79-x^88-x^91-x^94)*G(x^16) + (-1+x^16-x^48-x^60-x^63+2*x^64+x^76+x^79-x^80+x^112+x^124+x^127-x^128-x^140-x^143)*G(x^64) + (1-x^16-x^64+x^80-x^256+x^272+x^320-x^336)*G(x^256) = 0. - Robert Israel, May 18 2016

A141803 Triangle read by rows derived from generalized Thue-Morse sequences.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 3, 2, 0, 1, 2, 3, 1, 0, 0, 1, 2, 3, 4, 2, 2, 1, 1, 2, 3, 4, 1, 3, 0, 1, 1, 2, 3, 4, 5, 2, 0, 1, 0, 1, 2, 3, 4, 5, 1, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 1, 3, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 2, 4, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 3, 3, 0, 1

Offset: 1

Gary W. Adamson and Roger L. Bagula, Jul 06 2008

Triangle read by rows, antidiagonals of an array composed of generalized Thue-Morse sequences [defined in A010060, comment of Zizka]. For each row of the array, n>0; n-th term of m-th row (m>0) = sum of digits of n in base (m+1), mod (m+1).
Every row of the array starting from the n-th one as well as every row of the triangle starting from the (2*n-1)-th one begins from (1,2,3,...,n).
Row sums = A141804: (1, 2, 3, 5, 8, 7, 15, 15, 18, 22,...).
Row 1 of the array (corresponding to base 2) = A010060 (n>0), rows 2 - 8 are the sequences A053838 - A053844, row 9 = A053837.

			First few rows of the array are:
1, 1, 0, 1, 0, 0, 1, 1,...
1, 2, 1, 2, 0, 2, 0, 1,...
1, 2, 3, 1, 2, 3, 0, 2,...
1, 2, 3, 4, 1, 2, 3, 4,...
1, 2, 3, 4, 5, 1, 2, 3,...
1, 2, 3, 4, 5, 6, 1, 2,...
...
Triangle = antidiagonals of the array:
1;
1, 1;
1, 2, 0;
1, 2, 1, 1;
1, 2, 3, 2, 0;
1, 2, 3, 1, 0, 0;
1, 2, 3, 4, 2, 2, 1;
1, 2, 3, 4, 1, 3, 0, 1;
1, 2, 3, 4, 5, 2, 0, 1, 0;
1, 2, 3, 4, 5, 1, 3, 2, 1, 0;
1, 2, 3, 4, 5, 6, 2, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 1, 3, 0, 0, 0, 0;
1, 2, 3, 4, 5, 6, 7, 2, 4, 2, 1, 2, 1;
1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 3, 3, 0, 1;
...
a(8) = 2, = (3,2) of the array indicating that in the sequence 1,2,3,...mod 4, sum of digits of "2" mod 4 = 2.

Ivan Neretin, Table of n, a(n) for n = 1..8001

Cf. A010060, A053837 - A053844, A141804.

Flatten@Table[Mod[Total@IntegerDigits[n - i, i], i], {n, 16}, {i, n - 1, 2, -1}] (* Ivan Neretin, Jun 18 2018 *)

Explanation in the Comments section corrected by Andrey Zabolotskiy, May 18 2016

A053840 (Sum of digits of n written in base 5) modulo 5.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the fourth row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010874, A053824, A141803.

Cf. A010060, A053837-A053844.

Mod[Total@ IntegerDigits[#, 5], 5] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)

a(n) = vecsum(digits(n,5)) % 5; \\ Michel Marcus, May 16 2016

a(n) = A010874(A053824(n)). - Andrey Zabolotskiy, May 18 2016

A053843 (Sum of digits of n written in base 8) modulo 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 0, 2, 3, 4, 5, 6, 7, 0, 1, 3, 4, 5, 6, 7, 0, 1, 2, 4, 5, 6, 7, 0, 1, 2, 3, 5, 6, 7, 0, 1, 2, 3, 4, 6, 7, 0, 1, 2, 3, 4, 5, 7, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 0, 2, 3, 4, 5, 6, 7, 0, 1, 3, 4, 5, 6, 7, 0, 1, 2, 4, 5, 6, 7, 0, 1, 2, 3, 5, 6, 7, 0, 1, 2, 3, 4, 6

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the seventh row of the array in A141803. - Andrey Zabolotskiy, May 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010877, A053829, A141803.

Cf. A010060, A053837-A053844.

Table[Mod[Plus @@ IntegerDigits[n, 8], 8], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n) = A010877(A053829(n)). - Andrey Zabolotskiy, May 18 2016

A053841 (Sum of digits of n written in base 6) modulo 6.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the fifth row of the array in A141803. - Andrey Zabolotskiy, May 18 2016

Paolo Xausa, Table of n, a(n) for n = 0..10000
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010875, A053827, A141803.

Cf. A010060, A053837-A053844.

Mod[DigitSum[Range[0, 100], 6], 6] (* Paolo Xausa, Aug 09 2024 *)

a(n) = vecsum(digits(n, 6)) % 6; \\ Michel Marcus, May 18 2016

a(n) = A010875(A053827(n)). - Andrey Zabolotskiy, May 18 2016

A053842 (Sum of digits of n written in base 7) modulo 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 0, 2, 3, 4, 5, 6, 0, 1, 3, 4, 5, 6, 0, 1, 2, 4, 5, 6, 0, 1, 2, 3, 5, 6, 0, 1, 2, 3, 4, 6, 0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 0, 2, 3, 4, 5, 6, 0, 1, 3, 4, 5, 6, 0, 1, 2, 4, 5, 6, 0, 1, 2, 3, 5, 6, 0, 1, 2, 3, 4, 6, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 0, 1

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the sixth row of the array in A141803. - Andrey Zabolotskiy, May 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010876, A053828, A141803.

Cf. A010060, A053837-A053844.

Table[Mod[Plus @@ IntegerDigits[n, 7], 7], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n) = vecsum(digits(n, 7)) % 7; \\ Michel Marcus, May 18 2016

a(n) = A010876(A053828(n)). - Andrey Zabolotskiy, May 18 2016

cp's OEIS Frontend

A053838 a(n) = (sum of digits of n written in base 3) modulo 3.

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A053837 Sum of digits of n modulo 10.

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A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

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A141803 Triangle read by rows derived from generalized Thue-Morse sequences.

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A053840 (Sum of digits of n written in base 5) modulo 5.

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A053843 (Sum of digits of n written in base 8) modulo 8.

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A053841 (Sum of digits of n written in base 6) modulo 6.

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