A053735 -id:A053735 - cp's OEIS frontend

A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

0, 1, 2, 9, 10, 11, 18, 19, 20, 81, 82, 83, 90, 91, 92, 99, 100, 101, 162, 163, 164, 171, 172, 173, 180, 181, 182, 729, 730, 731, 738, 739, 740, 747, 748, 749, 810, 811, 812, 819, 820, 821, 828, 829, 830, 891, 892, 893, 900, 901, 902, 909, 910, 911

Offset: 0

Clark Kimberling

a(n) = Sum_{i=0..m} d(i)*9^i, where Sum_{i=0..m} d(i)*3^i is the base-3 representation of n.
Numbers that can be written using only digits 0, 1 and 2 in base 9. Also, write n in base 3, read as base 9: (3) [n] (9) in base change notation. a(3n+k) = 9a(n)+k for k in {0,1,2}. - Franklin T. Adams-Watters, Jul 24 2006
Also, every term k corresponds to a unique pair i,j with k = a(i) + 3*a(j) (similarly to the Moser-de Bruijn sequence). - Luis Rato, May 02 2024

Cf. A007089, A208665, A338086 (ternary digit duplication).

Cf. A053735, A053830.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 9
    end
r end
[a(n) for n in 0:53] |> println # Peter Luschny, Jan 03 2021

Table[FromDigits[RealDigits[n, 3], 9], {n, 1, 100}] (* Clark Kimberling, Aug 14 2012 *)
Select[Range[0,1000],Total[IntegerDigits[#,3]]==Total[IntegerDigits[#,9]]&] (* Harvey P. Dale, Feb 17 2020 *)

a(n) = {my(d = digits(n, 3)); subst(Pol(d), x, 9);} \\ Michel Marcus, Apr 09 2015

G.f. f(x) = Sum_{j>=0} 9^j*x^(3^j)*(1+x^(3^j)-2*x^(2*3^j))/((1-x)*(1-x^(3^(j+1)))) satisfies f(x) = 9*(x^2+x+1)*f(x^3) + x*(1+2*x)/(1-x^3). - Robert Israel, Apr 13 2015

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A230641 a(n) = n + (sum of digits in base-3 representation of n).

Original entry on oeis.org

0, 2, 4, 4, 6, 8, 8, 10, 12, 10, 12, 14, 14, 16, 18, 18, 20, 22, 20, 22, 24, 24, 26, 28, 28, 30, 32, 28, 30, 32, 32, 34, 36, 36, 38, 40, 38, 40, 42, 42, 44, 46, 46, 48, 50, 48, 50, 52, 52, 54, 56, 56, 58, 60, 56, 58, 60, 60, 62, 64, 64, 66, 68, 66, 68, 70, 70, 72, 74, 74, 76, 78, 76, 78, 80, 80, 82, 84, 84, 86, 88, 82

Offset: 0

N. J. A. Sloane, Oct 31 2013

The image of this sequence is the set of nonnegative even numbers (A005843). Joshi (1973) proved that the sequence of base-q self numbers (analogous to A003052) is the sequence of odd numbers (A005408) for all odd q. - Amiram Eldar, Nov 28 2020

V. S. Joshi, Ph.D. dissertation, Gujarat Univ., Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Donovan Johnson, Table of n, a(n) for n = 0..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)

Cf. A228088, A010061, A230091, A230092, A227915.

a230641 n = a053735 n + n  -- Reinhard Zumkeller, May 19 2015

Table[n + Plus @@ IntegerDigits[n, 3], {n, 0, 100}] (* Amiram Eldar, Nov 28 2020 *)

a(n) = n + A053735(n). - Amiram Eldar, Nov 28 2020

A065368 Alternating sum of ternary digits in n. Replace 3^k with (-1)^k in ternary expansion of n.

Original entry on oeis.org

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

Offset: 0

Marc LeBrun, Oct 31 2001

Notation: (3)[n](-1).

Fixed point of the morphism 0 -> 0,1,2; 1 -> -1,0,1; 2 -> -2,-1,0; ...; n -> -n,-n+1,-n+2. - Philippe Deléham, Oct 22 2011

			15 = +1(9)+2(3)+0(1) -> +1(+1)+2(-1)+0(+1) = -1 = a(15).

Cf. A030341, A053735, A065359, A065364.

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 3)[1:][::-1]))
print([a(n) for n in range(104)]) # Michael S. Branicky, Jul 28 2021

from sympy.ntheory import digits
def A065368(n): return sum((0, 1, 2, -1, 0, 1, -2, -1, 0)[i] for i in digits(n,9)[1:]) # Chai Wah Wu, Jul 19 2024

a(n) = Sum_{k>=0} A030341(n,k)*(-1)^k. - Philippe Deléham, Oct 22 2011.

G.f. A(x) satisfies: A(x) = x * (1 + 2*x) / (1 - x^3) - (1 + x + x^2) * A(x^3). - Ilya Gutkovskiy, Jul 28 2021

Initial 0 added by Philippe Deléham, Oct 22 2011

A230643 Number of integers m such that m + (sum of digits in base-3 representation of m) = 2n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 31 2013

Number of times 2n appears in A230641.

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

Cf. A230641, A230642.

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

A010063 a(n+1) = a(n) + sum of digits in base 3 representation of a(n), with a(0) = 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 18, 20, 24, 28, 30, 32, 36, 38, 42, 46, 50, 56, 60, 64, 68, 74, 80, 88, 92, 96, 100, 104, 110, 114, 118, 122, 128, 134, 142, 148, 154, 160, 168, 172, 176, 182, 188, 196, 202, 208, 214, 222, 228, 234, 240, 248, 252, 254, 258

Offset: 0

Leonid Broukhis

Donovan Johnson, Table of n, a(n) for n = 0..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,

NestList[#+Total[IntegerDigits[#,3]]&,1,60] (* Harvey P. Dale, Jun 14 2022 *)

More terms from Neven Juric, Apr 11 2008

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)

cp's OEIS Frontend

A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

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A230641 a(n) = n + (sum of digits in base-3 representation of n).

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A065368 Alternating sum of ternary digits in n. Replace 3^k with (-1)^k in ternary expansion of n.

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A230643 Number of integers m such that m + (sum of digits in base-3 representation of m) = 2n.

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A010063 a(n+1) = a(n) + sum of digits in base 3 representation of a(n), with a(0) = 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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A230854 Numbers n such that m + (sum of digits in base-3 representation of m) = n has exactly two solutions.

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