A053839 -id:A053839 - cp's OEIS frontend

A287552 Positions of 0 in A053839.

1, 8, 11, 14, 20, 23, 26, 29, 35, 38, 41, 48, 50, 53, 60, 63, 68, 71, 74, 77, 83, 86, 89, 96, 98, 101, 108, 111, 113, 120, 123, 126, 131, 134, 137, 144, 146, 149, 156, 159, 161, 168, 171, 174, 180, 183, 186, 189, 194, 197, 204, 207, 209, 216, 219, 222, 228

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 0, 1, 2, 3, with
in positions given by A287553,
in positions given by A287554,
in positions given by A287555,
in positions given by A287552.

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

Cf. A053839, A287553, A287554, A287555.

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];  (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A287555 Positions of 3 in A053839.

Original entry on oeis.org

4, 7, 10, 13, 19, 22, 25, 32, 34, 37, 44, 47, 49, 56, 59, 62, 67, 70, 73, 80, 82, 85, 92, 95, 97, 104, 107, 110, 116, 119, 122, 125, 130, 133, 140, 143, 145, 152, 155, 158, 164, 167, 170, 173, 179, 182, 185, 192, 193, 200, 203, 206, 212, 215, 218, 221, 227

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 1, 2, 3, 0, with
in positions given by A287552,
in positions given by A287553,
in positions given by A287554,
in positions given by A287555.

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

Cf. A053839, A287552, A287553, A287554.

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];         (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A287553 Positions of 1 in A053839.

Original entry on oeis.org

2, 5, 12, 15, 17, 24, 27, 30, 36, 39, 42, 45, 51, 54, 57, 64, 65, 72, 75, 78, 84, 87, 90, 93, 99, 102, 105, 112, 114, 117, 124, 127, 132, 135, 138, 141, 147, 150, 153, 160, 162, 165, 172, 175, 177, 184, 187, 190, 195, 198, 201, 208, 210, 213, 220, 223, 225

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, with
in positions given by A287554,
in positions given by A287555,
in positions given by A287552,
in positions given by A287553.

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

Cf. A053839, A287552, A287554, A287555.

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];         (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A287554 Positions of 2 in A053839.

Original entry on oeis.org

3, 6, 9, 16, 18, 21, 28, 31, 33, 40, 43, 46, 52, 55, 58, 61, 66, 69, 76, 79, 81, 88, 91, 94, 100, 103, 106, 109, 115, 118, 121, 128, 129, 136, 139, 142, 148, 151, 154, 157, 163, 166, 169, 176, 178, 181, 188, 191, 196, 199, 202, 205, 211, 214, 217, 224, 226

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, with
in positions given by A287555,
in positions given by A287552,
in positions given by A287553,
in positions given by A287554.

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

Cf. A053839, A287552, A287553, A287555.

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];  (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A273149 a(n) = A053839(n)+1.

Original entry on oeis.org

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

Offset: 0

William K. Grannis, May 16 2016

nonn

A four-way fair share sequence.  This is similar to the Thue-Morse Sequence. The Thue-Morse Sequence is the fairest way to split objects amongst two groups. If we call the groups A and B, most people split ABABABABABABABABABABABAB.......
This is unfair for B, because out of the best 2, A gets the best. Out of the second best 2, a gets the best. The Thue-Morse Sequence solves this:
ABBABAABBAABABBABAABABBAABBABAAB... The easiest way to generate the Thue-Morse Sequence is starting with a 1. Every 1 becomes 12. Every 2 becomes 21. Thus the sequence is obtained by recursion.
The present sequence is the same, but for splitting objects amongst 4 groups. Start with a 1. Every 1 becomes 1,2,3,4. Every 2 becomes 2,3,4,1. Every 3 becomes 3,4,1,2. Every 4 becomes 4,1,2,3.

Cf. A010060, A053839.

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

Original entry on oeis.org

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.

A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 31 2017

This is the fixed point of the morphism 0->0132, 1->1320, 2->3201, 3->2013 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.   Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1.
In the following guide to related sequences, column 1 indexes fixed points on {0,1,2,3}, and column 2 indicates position sequences of 0, 1, 2, 3.  Those sequences therefore comprise a 4-way splitting of the positive integers.
Fixed points of morphisms:                        Position sequences:
A053839: 0->0123, 1->1230, 2->2301, 3->3012       A287552-A287555
A287556: 0->0132, 1->1320, 2->3201, 3->2013       A287557-A287560
A287561: 0->0213, 1->2130, 2->1302, 3->3021       A287562-A287565
A287566: 0->0231, 1->2310, 2->3102, 3->1023       A287567-A287570
A287571: 0->0312, 1->3120, 2->1203, 3->2031       A287572-A287575
A287576: 0->0321, 1->3210, 2->2103, 3->1032       A287577-A287580

			First three iterations of the morphism:
0132
0132132020133201
0132132020133201132020133201013232010132132020132013320101321320

Clark Kimberling, Table of n, a(n) for n = 1..20000
Index entries for sequences that are fixed points of mappings

Cf. A287557, A287558, A287559, A287560.

s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9]; (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A079498 Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.

Original entry on oeis.org

0, 5, 7, 11, 13, 15, 19, 21, 26, 29, 31, 33, 37, 39, 44, 45, 50, 52, 55, 57, 62, 63, 68, 70, 74, 76, 78, 83, 85, 87, 91, 93, 98, 99, 104, 106, 109, 111, 116, 117, 122, 124, 128, 130, 132, 135, 140, 142, 146, 148, 150, 154, 156, 161, 163, 165, 170, 171, 176, 178, 182, 184

Offset: 1

Carlos Alves, Jan 21 2003

In base 2 this gives the "Evil Numbers" (cf. A001969). One may conjecture that in base b the asymptotic slope will be b and asymptotic density 1/b for each result (mod b). Cases b=31 or b=61 gave considerable number of primes on the sequence.
Proof of this conjecture: in general, the sequence d with terms d(n) = sum of digits of n written in base b (mod b) is a fixed point of the generalized Thue-Morse morphism 0->01..b-1, 1->12..0, etc. See A053839 for the case b=4. It follows directly from this that all symbols have asymptotic density 1/b, and therefore that the positional sequences all have asymptotic slope b. - Michel Dekking, Apr 18 2019
Positions of 0's in A053838. Cf. A026601.

			83 is a term since 83 = (1,0,0,0,2)_3 and 1 + 0 + 0 + 0 + 2 = 3 == 0 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969. See A053840 for base b=5. See A141803 for an array with all b.

Ev = Function[{b, x}, vx = IntegerDigits[x, b]; Mod[Apply[Plus, vx], b]]; Seq = Function[{b, n}, Flatten[Position[Table[Ev[b, k], {k, 0, n}], 0]] - 1]; sb = Seq[3, 1000]

a(1) = 0 inserted and offset corrected by Amiram Eldar, Jan 05 2020

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A287552 Positions of 0 in A053839.

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A287555 Positions of 3 in A053839.

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A287553 Positions of 1 in A053839.

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A287554 Positions of 2 in A053839.

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A273149 a(n) = A053839(n)+1.

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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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A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

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A079498 Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.

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