A286585 -id:A286585 - cp's OEIS frontend

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

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

A291759 Binary encoding of 2-digits in ternary representation of A048673(n).

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 5, 0, 3, 4, 1, 0, 1, 0, 1, 0, 5, 2, 9, 6, 7, 8, 5, 2, 7, 4, 1, 2, 1, 0, 1, 4, 3, 2, 1, 4, 1, 6, 9, 2, 3, 0, 17, 10, 13, 4, 13, 0, 23, 4, 9, 8, 5, 0, 13, 2, 9, 8, 1, 10, 3, 0, 1, 12, 3, 0, 1, 0, 11, 2, 5, 12, 5, 2, 1, 10, 9, 4, 1, 8, 11, 14, 17, 4, 5, 0, 5, 0, 15, 0, 33, 6, 21, 16, 25, 6, 11, 8, 25, 16, 3, 8, 45, 8, 9, 4, 17, 8

Offset: 1

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A048673, A286585, A286586, A289814, A291760, A291763.

a(n) = A289814(A048673(n)).

A286586 a(n) = A006047(A048673(n)).

Original entry on oeis.org

2, 3, 2, 6, 4, 9, 3, 12, 8, 6, 6, 18, 2, 18, 3, 24, 4, 12, 4, 12, 4, 9, 6, 36, 18, 27, 6, 36, 12, 54, 6, 48, 6, 6, 8, 24, 6, 18, 24, 24, 12, 6, 9, 18, 12, 36, 2, 72, 18, 27, 12, 54, 4, 81, 12, 72, 12, 18, 8, 108, 12, 9, 12, 96, 9, 36, 4, 12, 18, 36, 8, 48, 16, 27, 24, 36, 9, 36, 12, 48, 72, 18, 6, 12, 24, 54, 27, 36, 24, 18, 16, 72, 8, 81, 2

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000079, A006047, A042950, A048673, A286585, A286587.

from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
from functools import reduce
def a006047(n):
    d=digits(n, 3)
    return n + 1 if n<3 else reduce(mul, [1 + d[i] for i in range(1, len(d))])
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a006047(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A286586 n) (A006047 (A048673 n)))

a(n) = A006047(A048673(n)).

For n >= 0, a(A000079(n)) = A042950(n).

A286582 a(n) = A001222(A048673(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A048673, A091304, A278224, A286583, A286584, A286585.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A286582(n) = bigomega(A048673(n));

from sympy import factorint, nextprime, primefactors, prod
def a001222(n): return 0 if n==1 else a001222(n//primefactors(n)[-1]) + 1
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2
def a(n): return a001222(a048673(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 12 2017

(define (A286582 n) (A001222 (A048673 n)))

a(n) = A001222(A048673(n)).

A286583 a(n) = A007814(A048673(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007814, A048673, A286582, A286584, A286585.

Cf. A246261 (positions of zeros), A246263 (of nonzeros).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A007814(n) = (valuation(n,2));
A286583(n) = A007814(A048673(n));

from sympy import factorint, nextprime, prod
def a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2
def a(n): return a007814(a048673(n)) # Indranil Ghosh, Jun 12 2017

(define (A286583 n) (A007814 (A048673 n)))

a(n) = A007814(A048673(n)).

A286584 a(n) = A048673(n) mod 4.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A010873, A048673, A286582, A286583, A286585.

Cf. A246261 (positions of odd terms), A246263 (of even terms).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A286584(n) = (A048673(n)%4);

from sympy import factorint, nextprime
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return a048673(n)%4 # Indranil Ghosh, Jun 12 2017

(define (A286584 n) (modulo (A048673 n) 4))

a(n) = A010873(A048673(n)) = A048673(n) mod 4.

A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 03 2017

Reflecting the structure of A254103 also this sequence can be represented as a binary tree:
                                     0
                                     |
                  ...................1...................
                 2                                       1
       3......../ \........2                   4......../ \........2
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   4       1           3       3           5       3           5       2
  5 4     6 3         4 2     4 2         6 2     4 3         6 1     3 4
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A053735, A254103, A286585, A286633.

from sympy.ntheory.factor_ import digits
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n/2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))/2)
def a(n): return sum(digits(a254103(n), 3)[1:]) # Indranil Ghosh, Jun 06 2017

(define (A286632 n) (A053735 (A254103 n)))

a(n) = A053735(A254103(n)).
a(n) = A056239(A286633(n)).
For all n >= 0, a(A000079(n)) = n+1.

A340378 Number of 1-digits in the ternary representation of A048673(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

Binary weight of A304759(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A048673, A062756, A286585, A304759, A340379.

Cf. A340376 (positions of zeros).

PARI
```
A340378(n) = hammingweight(A304759(n));
```

a(n) = A062756(A048673(n)) = A000120(A304759(n)).

a(n) = A286585(n) - 2*A340379(n).

A340379 Number of 2-digits in the ternary representation of A048673(n).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 1, 2, 1, 3, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 2, 3, 1, 3, 0, 4, 1, 2, 1, 2, 0, 3, 1, 2, 1, 1, 2, 2, 0, 1, 2, 2, 0, 1, 0, 3, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 2, 0, 2, 0, 4, 0, 2, 2, 3, 1, 3, 2, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

Binary weight of A291759(n).

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

Cf. A000120, A048673, A081603, A286585, A291759, A292252.

Cf. A340377 (positions of zeros).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A340379(n) = hammingweight(A291759(n));

a(n) = A081603(A048673(n)) = A000120(A291759(n)).
a(n) = (A286585(n) - A340378(n)) / 2.
For all n >= 1, a(n) >= A292252(n).

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 2, 6, 7, 3, 3, 8, 1, 9, 2, 10, 11, 6, 4, 12, 11, 13, 3, 14, 9, 15, 3, 16, 12, 17, 3, 18, 3, 3, 7, 19, 3, 9, 19, 19, 6, 3, 5, 8, 12, 20, 1, 21, 8, 22, 12, 23, 11, 24, 12, 25, 6, 8, 26, 27, 12, 13, 12, 28, 13, 29, 4, 12, 9, 20, 26, 30, 31, 22, 10, 16, 5, 14, 6, 32, 33, 8, 3, 12, 10, 23, 15, 14, 19, 8

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A340381(n), A340382(n)], or equally, of the function f(n) = A290093(A048673(n)).

For all i, j: a(i) = a(j) => A286586(i) = A286586(j) => A286585(i) = A286585(j).

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

Cf. A048673, A278222, A290093, A291759, A304759, A340381, A340382, A340684.

Cf. also A290094, A305303.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
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; };
Aux340383(n) = [A278222(A291759(n)),A278222(A304759(n))];
v340383 = rgs_transform(vector(up_to,n,Aux340383(n)));
A340383(n) = v340383[n];

cp's OEIS Frontend

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

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A291759 Binary encoding of 2-digits in ternary representation of A048673(n).

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A286586 a(n) = A006047(A048673(n)).

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A286582 a(n) = A001222(A048673(n)).

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A286583 a(n) = A007814(A048673(n)).

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A286584 a(n) = A048673(n) mod 4.

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A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

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