A244042 -id:A244042 - cp's OEIS frontend

A300821 Möbius transform of A244042.

1, -1, 2, 4, 2, -2, 0, -4, 6, 8, 8, 8, 12, 12, 4, 10, 8, -6, 0, -14, 0, -4, 2, -8, -2, -12, 18, 12, 26, 16, 30, 20, 16, 20, 24, 24, 36, 36, 24, 44, 38, 24, 36, 28, 12, 26, 26, 20, 30, 22, 16, 24, 26, -18, -10, -24, 0, -22, 2, -28, 0, -30, 0, -20, -6, -8, 12, -20, 4, -36, 8, -24, 0, -36, -4, -36, -6, -24, 0, -50, 54, 44, 80, 24

Offset: 1

Antti Karttunen, Mar 14 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A244042, A300822, A300823.

Table[DivisorSum[n, MoebiusMu[n/#] FromDigits[IntegerDigits[#, 3] /. 2 -> 0, 3] &], {n, 84}] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300821(n) = sumdiv(n,d,moebius(n/d)*A244042(d));

a(n) = Sum_{d|n} moebius(n/d)*A244042(d).
a(n) = A000010(n) - A300822(n).
a(n) = A244042(n) - A300823(n).

A300823 Difference between A244042 and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 4, 3, 2, 1, 4, 1, 0, 5, 0, 1, 6, 1, 14, 3, 8, 1, 8, 3, 12, 9, 16, 1, 14, 1, 10, 11, 8, 3, 12, 1, 0, 15, -4, 1, 12, 1, 8, 15, 2, 1, 10, 1, 8, 11, 4, 1, 18, 11, 24, 3, 26, 1, 28, 1, 30, 9, 30, 15, 20, 1, 32, 5, 46, 1, 24, 1, 36, 7, 40, 9, 24, 1, 50, 27, 38, 1, 60, 11, 36, 29, 32, 1, 42, 13, 32, 33, 26, 3, 50, 1

Offset: 1

Antti Karttunen, Mar 14 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A051953, A244042, A300821, A300824, A300825.

f[n_] := FromDigits[IntegerDigits[n, 3] /. 2 -> 0, 3]; Table[f@ n - DivisorSum[n, MoebiusMu[n/#] f@ # &], {n, 97}] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300823(n) = -sumdiv(n,d,(dA244042(d));

a(n) = A244042(n) - A300821(n).
a(n) = -Sum_{d|n, dA008683(n/d)*A244042(d) = Sum_{d|n, dA300821(d).
a(n) = A051953(n) - A300824(n).

A359601 Dirichlet inverse of A244042, where A244042(n) replaces 2's with 0's in the ternary representation of n.

Original entry on oeis.org

1, 0, -3, -4, -3, 0, -1, 0, 0, -10, -9, 12, -13, -12, 9, 6, -9, 0, -1, 24, 3, -4, -3, 0, 8, 0, 0, -20, -27, 30, -31, -30, 27, -28, -21, 0, -37, -36, 39, 40, -39, 36, -37, 36, 0, -28, -27, -18, -30, 30, 27, 76, -27, 0, 53, 96, 3, -4, -3, -72, -1, 0, 0, 6, 69, 12, -13, 60, 9, 82, -9, 0, -1, 0, -24, 4, 15

Offset: 1

Antti Karttunen, Jan 11 2023

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A244042, A300821, A359602.

Cf. A056911 (positions of odd terms), A323239 (parity of terms), A337945.

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
memoA359601 = Map();
A359601(n) = if(1==n,1,my(v); if(mapisdefined(memoA359601,n,&v), v, v = -sumdiv(n,d,if(dA244042(n/d)*A359601(d),0)); mapput(memoA359601,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA244042(n/d) * a(d).

a(n) = A359602(n) - A244042(n).

A359602 Sum of A244042 and its Dirichlet inverse, where A244042(n) replaces 2's with 0's in the ternary representation of n.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 24, 0, 0, 18, 16, 0, 0, 0, 24, 6, 0, 0, 0, 9, 0, 27, 8, 0, 60, 0, 0, 54, 0, 6, 36, 0, 0, 78, 80, 0, 72, 0, 72, 27, 0, 0, 12, 1, 60, 54, 104, 0, 0, 54, 96, 6, 0, 0, -72, 0, 0, 9, 16, 78, 24, 0, 72, 18, 92, 0, 0, 0, 0, -21, 8, 18, 0, 0, -84, 81, 0, 0, 144, 54, 0, 162, 32

Offset: 1

Antti Karttunen, Jan 11 2023

Antti Karttunen, Table of n, a(n) for n = 1..19682

Cf. A244042, A359601.

Cf. A053850 (positions of odd terms), A353569 (parity of terms).

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
memoA359601 = Map();
A359601(n) = if(1==n,1,my(v); if(mapisdefined(memoA359601,n,&v), v, v = -sumdiv(n,d,if(dA244042(n/d)*A359601(d),0)); mapput(memoA359601,n,v); (v)));
A359602(n) = (A244042(n)+A359601(n));

a(n) = A244042(n) + A359601(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A244042(d) * A359601(n/d).

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

A300222 In ternary (base-3) representation of n, replace 1's with 0's.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 54, 54, 56, 54, 54, 56, 60, 60, 62, 54, 54, 56, 54, 54, 56, 60, 60, 62, 72, 72, 74, 72, 72, 74, 78, 78, 80, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2

Offset: 0

Antti Karttunen, Mar 14 2018

			For n=46, which in base-3 (A007089) is 1201, replacing 1's with 0's gives 200, and as that is base-3 representation of 18 (= 2*(3^2) + 0*(3^1) + 0*(3^0)), a(46) = 18.

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A004488, A005836, A007089, A244042, A289814.

Cf. A300822 (Moebius transform).

Array[FromDigits[IntegerDigits[#, 3] /. 1 -> 0, 3] &, 93, 0] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300222(n) = (n - A244042(n));
\\ Or directly as:
A300222(n) = fromdigits(apply(x->(if (1==x, 0, x)), digits(n, 3)), 3);

a(n) = n - A244042(n) = 2*A244042(A004488(n)).
a(n) = 2*A005836(1+A289814(n)). [With the current starting offset 1 of A005836.]
a(n) = A300822(n) + A300824(n).

A300825 Filter sequence combining A300823(n) and A300824(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 9, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 44, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 64, 67, 68, 69

Offset: 1

Antti Karttunen, Mar 14 2018

nonn

Restricted growth sequence transform of ordered pair [A300823(n), A300824(n)].

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A051953, A244042, A300222, A300823, A300824.

Cf. also A293226, A300833.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300823(n) = -sumdiv(n,d,(dA244042(d));
A300222(n) = (n - A244042(n));
A300824(n) = -sumdiv(n,d,(dA300222(d));
Aux300825(n) = [A300823(n), A300824(n)];
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300825(n))),"b300825.txt");

cp's OEIS Frontend

A300821 Möbius transform of A244042.

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A300823 Difference between A244042 and its Möbius transform.

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A359601 Dirichlet inverse of A244042, where A244042(n) replaces 2's with 0's in the ternary representation of n.

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A359602 Sum of A244042 and its Dirichlet inverse, where A244042(n) replaces 2's with 0's in the ternary representation of n.

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A004488 Tersum n + n.

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

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A300222 In ternary (base-3) representation of n, replace 1's with 0's.

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