A289814 -id:A289814 - cp's OEIS frontend

A293222 a(n) = Product_{d|n, dA019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of proper divisors of n.

1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 6, 2, 12, 1, 6, 1, 4, 3, 4, 1, 36, 2, 2, 1, 12, 1, 36, 1, 36, 2, 12, 6, 30, 1, 10, 1, 240, 1, 180, 1, 20, 6, 20, 1, 1620, 3, 60, 6, 60, 1, 30, 4, 72, 5, 4, 1, 360, 1, 2, 15, 72, 2, 180, 1, 36, 10, 144, 1, 2700, 1, 2, 90, 20, 6, 180, 1, 720, 1, 4, 1, 540, 12, 6, 2, 720, 1, 900, 3, 100, 1, 20, 10, 16200, 1, 60, 6

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..6561

Cf. A019565, A289814, A293221, A293224 (restricted growth sequence transform), A293226.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Remy Sigrist
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };

a(n) = Product_{d|n, dA019565(A289814(d)).

A289831 a(n) = A289813(n) + A289814(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 2, 3, 3, 4, 5, 5, 6, 7, 7, 6, 7, 7, 4, 5, 5, 6, 7, 7, 6, 7, 7, 8, 9, 9, 10, 11, 11, 10, 11, 11, 12, 13, 13, 14, 15, 15, 14, 15, 15, 12, 13, 13, 14, 15, 15, 14, 15, 15, 8, 9, 9, 10, 11, 11, 10, 11, 11, 12, 13, 13, 14, 15, 15, 14, 15, 15, 12

Offset: 0

Rémy Sigrist, Jul 13 2017

The ones in the binary representation of a(n) correspond to the nonzero digits in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 1110 (a(42) = 14).
Each number k >= 0 appears 2^A000120(k) times.
a(A004488(n)) = a(n).

			The first values, alongside the ternary representation of n, and the binary representation of a(n), are:
n       a(n)    ternary(n)  binary(a(n))
--      ----    ----------  ------------
0       0       0           0
1       1       1           1
2       1       2           1
3       2       10          10
4       3       11          11
5       3       12          11
6       2       20          10
7       3       21          11
8       3       22          11
9       4       100         100
10      5       101         101
11      5       102         101
12      6       110         110
13      7       111         111
14      7       112         111
15      6       120         110
16      7       121         111
17      7       122         111
18      4       200         100
19      5       201         101
20      5       202         101
21      6       210         110
22      7       211         111
23      7       212         111
24      6       220         110
25      7       221         111
26      7       222         111

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A000120, A004488, A289813, A289814.

Table[FromDigits[Sign@ IntegerDigits[n, 3], 2], {n, 0, 100}] (* Indranil Ghosh, Aug 03 2017 *)

a(n) = my (d=digits(n,3)); fromdigits(vector(#d, i, sign(d[i])), 2)

from sympy.ntheory.factor_ import digits
from sympy import sign
def a(n):
    d=digits(n, 3)[1:]
    return int(''.join(str(sign(i)) for i in d), 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 03 2017

a(0) = 0.
a(3*n) = 2*a(n).
a(3*n + 1) = 2*a(n) + 1.
a(3*n + 2) = 2*a(n) + 1.

A290092 Filter based on 2-digits of base-3 expansion: a(n) = A278222(A289814(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 1, 1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 4, 4, 8, 1, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 1, 1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 4, 4, 8, 2, 2, 6, 2, 2, 6, 6, 6, 12, 2, 2, 6, 2, 2, 6, 6, 6, 12, 4, 4, 12, 4, 4, 12, 8, 8, 16, 1, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 1, 1, 2, 2, 2, 4, 2, 2, 6, 2, 2, 6, 4, 4, 8, 1, 1, 2, 1

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A278222, A289814, A290091, A290093.

(define (A290092 n) (A278222 (A289814 n)))

a(n) = A278222(A289814(n)).

A351092 a(n) = Product_{d|A000265(n)} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the odd divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 6, 1, 6, 1, 5, 2, 15, 2, 10, 1, 30, 1, 1, 3, 2, 6, 1, 1, 6, 6, 36, 1, 1, 5, 1, 2, 2, 15, 3, 2, 30, 10, 10, 1, 15, 30, 90, 1, 30, 1, 28, 3, 35, 2, 14, 6, 21, 1, 105, 1, 28, 6, 7, 6, 210, 36, 42, 1, 35, 1, 3150, 5, 420, 1, 105, 2, 1, 2, 2, 15, 12, 3, 6, 2, 6, 30, 3, 10, 1, 10

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A000265, A019565, A289814, A351082, A351091, A351094 (rgs-transform).

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };

a(n) = A351082(A000265(n)).

A294932 Multiplicative with a(p^e) = A019565(A289814(e)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A019565, A289814, A293442, A294931, A294933.

(define (A294932 n) (if (= 1 n) n (* (A019565 (A289814 (A067029 n))) (A294932 (A028234 n)))))

A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 12, 1, 2, 6, 1, 1, 12, 1, 1, 12, 3, 1, 12, 1, 1, 2, 15, 3, 216, 1, 5, 2, 6, 1, 6, 1, 2, 36, 5, 1, 180, 3, 10, 30, 1, 1, 1080, 1, 3, 10, 1, 1, 3240, 1, 1, 36, 1, 1, 20, 1, 450, 10, 30, 1, 45360, 1, 1, 30, 75, 3, 10, 1, 60, 360, 1, 1, 540, 15, 105, 2, 2, 1, 3240, 3, 50, 2, 35, 5, 2520, 1, 630, 60, 90, 1, 900

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Used as a part of filter A296073.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879, A019565, A289814, A295882, A296071, A296073.

Cf. also A293222.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296072(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(A295882(d))))); m; };

(define (A296072 n) (let loop ((m 1) (props (proper-divisors n))) (cond ((null? props) m) (else (loop (* m (A019565 (A289814 (A295882 (car props))))) (cdr props))))))
(define (proper-divisors n) (reverse (cdr (reverse (divisors n)))))
(define (divisors n) (let loop ((k n) (divs (list))) (cond ((zero? k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))

a(n) = Product_{d|n, dA019565(A289814(A295882(d))).

A351082 a(n) = Product_{d|n} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 2, 6, 3, 12, 1, 4, 2, 6, 1, 12, 6, 36, 6, 30, 5, 40, 15, 20, 10, 540, 30, 60, 1, 12, 2, 36, 1, 72, 6, 36, 36, 30, 1, 20, 1, 240, 2, 540, 3, 120, 30, 100, 10, 8100, 15, 600, 90, 900, 30, 210, 28, 1008, 35, 28, 14, 7560, 21, 84, 105, 504, 28, 1260, 7, 504, 210, 3024, 42, 94500, 35, 140, 3150, 700, 420, 18900

Offset: 1

Antti Karttunen, Jan 31 2022

Cf. A019565, A289814, A293221, A351081, A351092.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A351082(n) = { my(m=1); fordiv(n,d,m *= A019565(A289814(d))); (m); };

a(n) = A019565(A289814(n)) * A293222(n).

A053735 Sum of digits of (n written in base 3).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2}, 1->{1,2,3}, 2->{2,3,4}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 2 + 0 + 2 = 4 because 20 is written as 202 base 3.
From _Omar E. Pol_, Feb 20 2010: (Start)
This can be written as a triangle with row lengths A025192 (see the example in the entry A000120):
0,
1,2,
1,2,3,2,3,4,
1,2,3,2,3,4,3,4,5,2,3,4,3,4,5,4,5,6,
1,2,3,2,3,4,3,4,5,2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,2,3,4,3,4,5,4,5,6,3,...
where the k-th row contains a(3^k+i) for 0<=i<2*3^k and converges to A173523 as k->infinity. (End) [Changed conjectures to statements in this entry. - _Franklin T. Adams-Watters_, Jul 02 2015]
G.f. = x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + x^9 + 2*x^10 + ...

T. D. Noe, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS, Vol. 12 (2009), Article 09.5.6.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Michael Gilleland, Some Self-Similar Integer Sequences.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See pp. 11, 13.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Math. Semesterber, Vol. 49 (2002), pp. 209-226; preprint.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. p.205).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A065363, A007089, A173523. See A134451 for iterations.
Cf. A003137, A138530.
Sum of digits of n written in bases 2-16: A000120, this sequence, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-3 sequences: A006047, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1), A286585, A286632, A289813, A289814.

a053735 = sum . a030341_row
-- Reinhard Zumkeller, Feb 21 2013, Feb 19 2012

m=1; for u=0:104; sol(m)=sum(dec2base(u,3)-'0'); m=m+1;end
sol; % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,3):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

seq(convert(convert(n,base,3),`+`),n=0..100); # Robert Israel, Jul 02 2015

Table[Plus @@ IntegerDigits[n, 3], {n, 0, 100}] (* or *)
Nest[Join[#, # + 1, # + 2] &, {0}, 6] (* Robert G. Wilson v, Jul 27 2006 and modified Jul 27 2014 *)

{a(n) = if( n<1, 0, a(n\3) + n%3)}; /* Michael Somos, Mar 06 2004 */

A053735(n)=sumdigits(n,3) \\ Requires version >= 2.7. Use sum(i=1,#n=digits(n,3),n[i]) in older versions. - M. F. Hasler, Mar 15 2016

(define (A053735 n) (let loop ((n n) (s 0)) (if (zero? n) s (let ((d (mod n 3))) (loop (/ (- n d) 3) (+ s d)))))) ;; For R6RS standard. Use modulo instead of mod in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Jun 03 2017

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(3n) = a(n), a(3n + 1) = a(n) + 1, a(3n + 2) = a(n) + 2.
a(n) = n - 2*Sum_{k>0} floor(n/3^k) = n - 2*A054861(n). (End)
a(n) = A062756(n) + 2*A081603(n). - Reinhard Zumkeller, Mar 23 2003
G.f.: (Sum_{k >= 0} (x^(3^k) + 2*x^(2*3^k))/(1 + x^(3^k) + x^(2*3^k)))/(1 - x). - Michael Somos, Mar 06 2004, corrected by Franklin T. Adams-Watters, Nov 03 2005
In general, the sum of digits of (n written in base b) has generating function (Sum_{k>=0} (Sum_{0 <= i < b} i*x^(i*b^k))/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
First differences of A094345. - Vladeta Jovovic, Nov 08 2005
a(A062318(n)) = n and a(m) < n for m < A062318(n). - Reinhard Zumkeller, Feb 26 2008
a(n) = A138530(n,3) for n > 2. - Reinhard Zumkeller, Mar 26 2008
a(n) <= 2*log_3(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = Sum_{k>=0} A030341(n, k). - Philippe Deléham, Oct 21 2011
G.f. satisfies G(x) = (x+2*x^2)/(1-x^3) + (1+x+x^2)*G(x^3), and has a natural boundary at |x|=1. - Robert Israel, Jul 02 2015
a(n) = A056239(A006047(n)). - Antti Karttunen, Jun 03 2017
a(n) = A000120(A289813(n)) + 2*A000120(A289814(n)). - Antti Karttunen, Jul 20 2017
a(0) = 0; a(n) = a(n - 3^floor(log_3(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 3*log(3)/2 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

Original entry on oeis.org

0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, -8, -10, -6, -5, -7, -12, -11, -13, 27, 28, 26, 30, 31, 29, 24, 25, 23, 36, 37, 35, 39, 40, 38, 33, 34, 32, 18, 19, 17, 21, 22, 20, 15, 16, 14, -27, -26, -28, -24, -23, -25, -30, -29, -31, -18, -17, -19, -15, -14, -16, -21, -20, -22, -36

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

As the graph demonstrates, there are large discontinuities in the sequence between terms 3^i-1 and 3^i, and between terms 2*3^i-1 and 2*3^i. - N. J. A. Sloane, Jul 03 2016

			7 in base 3 is 21; changing the 2 to a (-1) gives (-1)*3+1 = -2, so a(7) = -2. I.e., the number of -2 according to the balanced ternary enumeration is 7, which can be obtained by replacing every -1 by 2 in the balanced ternary representation (or expansion) of -2, which is -1,1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 729 terms from A. Karttunen)
Ken Levasseur, The Balanced Ternary Number System
Tilman Piesk, First 27 numbers with their ternary representation
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Wikipedia, Balanced ternary

Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.

Column k=1 of A319047.

f:= proc(n) local L,i;
   L:= subs(2=-1,convert(n,base,3));
   add(L[i]*3^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]);
# alternate:
N:= 100: # to get a(0) to a(N)
g:= 0:
for n from 1 to ceil(log[3](N+1)) do
g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
od:
seq(coeff(g,x,j),j=0..N); # Robert Israel, Nov 17 2015
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Aug 14 2019

Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* Michael De Vlieger, Nov 17 2015 *)

a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
vector(73, i, a(i-1))  \\ Gheorghe Coserea, Nov 17 2015

def a(n):
    if n==0: return 0
    if n%3==0: return 3*a(n//3)
    elif n%3==1: return 3*a((n - 1)//3) + 1
    else: return 3*a((n - 2)//3) - 1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 06 2017

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
G.f. satisfies A(x) = 3*A(x^3)*(1+x+x^2) + x/(1+x+x^2). - corrected by Robert Israel, Nov 17 2015
A004488(n) = a(n)^{-1}(-a(n)). I.e., if a(n) <= 0, A004488(n) = A117967(-a(n)) and if a(n) > 0, A004488(n) = A117968(a(n)).
a(n) = n - 3 * A005836(A289814(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

Name corrected by Andrey Zabolotskiy, Nov 10 2019

cp's OEIS Frontend

A293222 a(n) = Product_{d|n, dA019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of proper divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A289831 a(n) = A289813(n) + A289814(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A290092 Filter based on 2-digits of base-3 expansion: a(n) = A278222(A289814(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A351092 a(n) = Product_{d|A000265(n)} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the odd divisors of n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A294932 Multiplicative with a(p^e) = A019565(A289814(e)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

A296072 a(n) = Product_{d|n, dA019565(A289814(A295882(d))); a product obtained from the -1's present in balanced ternary representation of the deficiencies of the proper divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

A351082 a(n) = Product_{d|n} A019565(A289814(d)); a product obtained from the 2-digits present in ternary expansions of the divisors of n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A053735 Sum of digits of (n written in base 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

MATLAB

Magma

Maple