A134451 -id:A134451 - cp's OEIS frontend

A160390 Decimal expansion of sqrt(3) - 1.

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

Offset: 0

Harry J. Smith, May 31 2009

Continued fraction expansion leads to the ternary digital root of n.

			0.732050807568877293527446341505872366942805253810380628055806979451933...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for algebraic numbers, degree 2

Cf. A002194, A134451 (continued fraction).

SetDefaultRealField(RealField(100)); Sqrt(3) -1; // G. C. Greubel, Nov 20 2018

RealDigits[Sqrt[3]-1,10,120][[1]] (* Harvey P. Dale, Dec 16 2016 *)

default(realprecision, 20080); x=10*(sqrt(3)-1); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b160390.txt", n, " ", d));

numerical_approx(sqrt(3) -1, digits=100) # G. C. Greubel, Nov 20 2018

A230639 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives B(n).

Original entry on oeis.org

1, 3, 5, 17, 29, 139, 249, 64570209, 129140169, 34315253252541, 68630377364913, 1044297913696328396542704032390321722034449074468444246791788357605, 2088595827392656793085408064780643444068898148936888424953199350297

Offset: 2

N. J. A. Sloane, Oct 31 2013

The largest power of 3 in M(n) = A230640(n).

Max Alekseyev, Table n, expression for a(n) for n=2..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Index entries for Colombian or self numbers and related sequences

Cf. A230093, A230640 (for M(n)).

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0, 0]);
else
B:=(f(ceil(n/2))[2] + f(floor(n/2))[2] + 2)/2;
M:=3^B+f(floor(n/2))[2]+1; RETURN([B, M]); fi;
end proc;
[seq(f(n)[1], n=1..9)];

Terms a(10) onward from Max Alekseyev, Nov 02 2013

A230642 Number of integers m such that m + (sum of digits in base-3 representation of m) = n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 31 2013

The usual convention in the OEIS is to omit the zero terms when every second term is zero. An exception was made in this case in order to preserve the parallels with A228085 and A230632. See also A230663.

a(n) is the number of times n occurs in A230641.

Donovan Johnson, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A230632, A228085, A230643, A230641.

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

A230853 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly one solution.

Original entry on oeis.org

0, 2, 6, 16, 26, 34, 44, 54, 62, 72, 98, 108, 116, 126, 136, 144, 154, 180, 190, 198, 208, 218, 226, 236, 260, 270, 278, 288, 298, 306, 316, 342, 352, 360, 370, 380, 388, 398, 424, 434, 442, 452, 462, 470, 480, 504, 514, 522, 532, 542, 550, 560, 586, 596, 604, 614, 624, 632, 642, 668, 678, 686

Offset: 1

N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

Select[Tally[Table[m+Total[IntegerDigits[m,3]],{m,0,700}]],#[[2]]==1&][[;;,1]] (* Harvey P. Dale, Feb 13 2023 *)

A230854 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly two solutions.

Original entry on oeis.org

4, 8, 10, 12, 14, 18, 20, 22, 24, 30, 36, 38, 40, 42, 46, 48, 50, 52, 58, 64, 66, 68, 70, 74, 76, 78, 80, 82, 88, 90, 92, 94, 96, 100, 102, 104, 106, 112, 118, 120, 122, 124, 128, 130, 132, 134, 140, 146, 148, 150, 152, 156, 158, 160, 162, 164, 170, 172, 174, 176, 178, 182, 184, 186, 188

Offset: 1

N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

Select[Tally[Table[m+Total[IntegerDigits[m,3]],{m,200}]],#[[2]]==2&][[All,1]] (* Harvey P. Dale, Aug 17 2019 *)

A230855 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly three solutions.

Original entry on oeis.org

28, 32, 56, 60, 84, 86, 110, 114, 138, 142, 166, 168, 192, 196, 220, 224, 244, 252, 272, 276, 300, 304, 328, 330, 354, 358, 382, 386, 410, 412, 436, 440, 464, 468, 488, 496, 516, 520, 544, 548, 572, 574, 598, 602, 626, 630, 654, 656, 680, 684, 708, 712, 730, 732, 734, 736, 738, 740, 758, 762

Offset: 1

N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

A230856 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly four solutions.

Original entry on oeis.org

248, 492, 978, 1222, 1708, 1952, 2192, 2196, 2436, 2680, 3166, 3410, 3896, 4140, 4380, 4384, 4624, 4868, 5354, 5598, 6084, 6328, 6566, 6572, 6810, 7054, 7540, 7784, 8270, 8514, 8754, 8758, 8998, 9242, 9728, 9972, 10458, 10702, 10942, 10946, 11186, 11430, 11916, 12160, 12646, 12890, 13128

Offset: 1

N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

A134452 Balanced ternary digital root of n.

Original entry on oeis.org

0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

Reinhard Zumkeller, Oct 27 2007

a(A005843(n))=0; a(A134453(n))=-1; a(A134454(n))=1; abs(a(A005408(n)))=1;

abs(a(n)) = A000035(n).

			42 == '+---0' --> +1-1-2-1+0=-2 == '-+' --> -1+1=0;
43 == '+---+' --> +1-1-2-1+1=-1;

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

R. Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digital Root
Wikipedia, Balanced Ternary

Cf. A065363, A134453, A134454, A005843, A005408, A000035.

Cf. A134451.

a(n) = f(n) where f(n) = if n<-1 then f(-A065363(-n)) else (if n>1 then f(A065363(n)) else n).

A382488 The number of unitary 3-smooth divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 29 2025

Period 6: repeat [1, 2, 2, 2, 1, 4].
Decimal expansion of 407380/333333.
Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).

Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten

a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];

Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.
a(n) = A034444(A065331(n)).
a(n) = A034444(n) if and only if n is 3-smooth (A003586).
a(n) = A072078(n) if and only if n is squarefree (A005117).
a(n) = abs(A181982(n+9)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.
G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * zeta(s).

A288699 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 494", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

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

Offset: 0

Robert Price, Jun 13 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A134451, A288697, A288697, A288700.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 494; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

cp's OEIS Frontend

A160390 Decimal expansion of sqrt(3) - 1.

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A230639 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives B(n).

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A230642 Number of integers m such that m + (sum of digits in base-3 representation of m) = n.

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A230853 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly one solution.

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Mathematica

A230854 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly two solutions.

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A230855 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly three solutions.

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A230856 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly four solutions.

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A134452 Balanced ternary digital root of n.

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A382488 The number of unitary 3-smooth divisors of n.

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