A053735 -id:A053735 - cp's OEIS frontend

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

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

Offset: 0

Clark Kimberling

			Triangle begins :
0
1
2
0, 1
1, 1
2, 1
0, 2
1, 2
2, 2
0, 0, 1
1, 0, 1
2, 0, 1
0, 1, 1
1, 1, 1
2, 1, 1 ...

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A081604 (row lengths), A053735 (row sums), A007089, A003137.

Cf. A030308, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

a030341 n k = a030341_tabf !! n !! k
a030341_row n = a030341_tabf !! n
a030341_tabf = iterate succ [0] where
   succ []     = [1]
   succ (2:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Feb 21 2013

A030341_row := n -> op(convert(n, base, 3)):
seq(A030341_row(n), n=0..32); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,3]],{n,0,40}]] (* Harvey P. Dale, Oct 20 2014 *)

A030341(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\3^k%3 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.

Original entry on oeis.org

0, 1, 2, 5, 8, 17, 26, 53, 80, 161, 242, 485, 728, 1457, 2186, 4373, 6560, 13121, 19682, 39365, 59048, 118097, 177146, 354293, 531440, 1062881, 1594322, 3188645, 4782968, 9565937, 14348906, 28697813, 43046720, 86093441, 129140162

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001

WARNING: The offset of this sequence has been changed from 0 to 1 without correcting the formulas and programs, many of them correspond to the original indexing a(0)=0, a(1)=1, ... - M. F. Hasler, Oct 06 2014
Numbers n such that no entry in n-th row of Pascal's triangle is divisible by 3, i.e., such that A062296(n) = 0.
The base 3 representation of these numbers is 222...222 or 122...222.
a(n+1) is the smallest number with ternary digit sum = n: A053735(a(n+1)) = n and A053735(m) <> n for m < a(n+1). - Reinhard Zumkeller, Sep 15 2006
A138002(a(n)) = 0. - Reinhard Zumkeller, Feb 26 2008
Also, number of terms in S(n), where S(n) is defined in A114482. - N. J. A. Sloane, Nov 13 2014
a(n+1) is also the Moore lower bound on the order of a (4,g)-cage. - Jason Kimberley, Oct 30 2011

			The first rows in Pascal's triangle with no multiples of 3 are:
row 0: 1;
row 1: 1, 1;
row 2: 1, 2,  1;
row 5: 1, 5, 10, 10,  5,  1;
row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1;

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023. See p. 9.
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A062296, A024023, A048473, A114482. Pairwise sums of A052993.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), this sequence (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A037233 (actual order of a (4,g)-cage).
Smallest number whose base b sum of digits is n: A000225 (b=2), this sequence (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10).

I:=[0,1,2]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2) -3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

A062318 :=proc(n)
    if n mod 2 = 1 then
        3^((n-1)/2)-1
    else
        2*3^(n/2-1)-1
    fi
end proc:
seq(A062318(n), n=1..37); # Emeric Deutsch, Feb 03 2005, offset updated

CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)),{x,0,40}],x] (* Vincenzo Librandi, Apr 20 2012 *)
A062318[n_]:= (1/3)*(Boole[n==0] -3 +3^(n/2)*(2*Mod[n+1,2] +Sqrt[3] *Mod[n, 2]));
Table[A062318[n], {n, 50}] (* G. C. Greubel, Apr 17 2023 *)

PARI
```
a(n)=3^(n\2)<M. F. Hasler, Oct 06 2014
```

def A062318(n): return (1/3)*(int(n==0) - 3 + 2*((n+1)%2)*3^(n/2) + (n%2)*3^((n+1)/2))
[A062318(n) for n in range(1,41)] # G. C. Greubel, Apr 17 2023

a(n) = 2*3^(n/2-1)-1 if n is even; a(n) = 3^(n/2-1/2)-1 if n is odd. - Emeric Deutsch, Feb 03 2005, offset updated.
From Paul Curtz, Feb 21 2008: (Start)
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3).
Partial sums of A108411. (End)
G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - Colin Barker, Apr 02 2012
a(2n+1) = 3*a(2n-1) + 2;  a(2n) = ( a(2n-1) + a(2n+1) )/2. See A060647 for case where a(1)= 1. - Richard R. Forberg, Nov 30 2013
a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - Luce ETIENNE, Aug 29 2014
a(n) = A052993(n-1) + A052993(n-2). - R. J. Mathar, Sep 10 2021
E.g.f.: (1 - 3*cosh(x) + 2*cosh(sqrt(3)*x) - 3*sinh(x) + sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Apr 06 2022
a(n) = (1/3)*([n=0] - 3 + (1+(-1)^n)*3^(n/2) + ((1-(-1)^n)/2)*3^((n+1)/2)). - G. C. Greubel, Apr 17 2023

More terms from Emeric Deutsch, Feb 03 2005

Entry revised by N. J. A. Sloane, Jul 29 2011

A051064 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Gary W. Adamson

a(n) is the Hamming distance between n and n-1 in ternary representation. - Philippe Deléham, Mar 29 2004
3^a(n) divides 4^n-1. - Benoit Cloitre, Oct 25 2004
Generalized Ruler Function for k=3. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
a(A007417(n)) is odd and a(A145204(n)) is even. - Reinhard Zumkeller, May 23 2013
First n terms comprise least cubefree word of length n using positive integers, where "cubefree" means that the word contains no three consecutive identical subwords; e.g., 1 contains no cube; 11 contains no cube; 111 does but 112 does not; ... 1,1,2,1,1,2,1,1,1 does, and 1,1,2,1,1,2,1,1,2 does, but 1,1,2,1,1,2,1,1,3 does not, etc. - Clark Kimberling, Sep 10 2013
The sequence is invariant under the "lower trim" operator: remove all ones, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017
a(n) is the dimension in which the coordinates of the vertices n-1 and n differ in the ternary reflected Gray code. - Arie Bos, Jul 12 2023
The number of powers of 3 that divide n. - Amiram Eldar, Mar 29 2025

			3^2 | 3*6 = 18, so a(6) = 2.

Letter from Gary W. Adamson to N. J. A. Sloane concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 243. Book's website
Tamas Lengyel, Divisiblity Properties by Multisection, Fib. Quart. 41 (1) (2003) 72.
Simon Plouffe, On the values of the functions zeta and gamma, arXiv preprint arXiv:1310.7195 [math.NT], 2013.
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=3.)
Index entries for sequences that are fixed points of mappings.

Cf. A007949.
Partial sums give A004128.
Cf. A000005, A027746.
Cf. A254046.
Cf. A001511, A055457, A115362, A373216, A373217.

a051064 = (+ 1) . length .
                  takeWhile (== 3) . dropWhile (== 2) . a027746_row
-- Reinhard Zumkeller, May 23 2013

seq(1+padic:-ordp(n,3), n=1..100); # Robert Israel, Aug 07 2014

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 1, 2}, 2 -> {1, 1, 3}, 3 -> {1, 1, 4}, 4 -> {1, 1, 5}})]}], {1}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
Table[ IntegerExponent[3n, 3], {n, 1, 105}] (* Jean-François Alcover, Oct 10 2011 *)

PARI
```
a(n)=if(n<1,0,1+valuation(n,3))
```

def A051064(n):
    c = 1
    a, b = divmod(n,3)
    while b == 0:
        a, b = divmod(a,3)
        c += 1
    return c # Chai Wah Wu, Apr 18 2022

a(n) = A007949(n) + 1 = A004128(n) - A004128(n-1).
Multiplicative with a(p^e) = e+1 if p = 3; 1 if p <> 3. - Vladeta Jovovic, Aug 24 2002
G.f.: Sum_{k>=0} x^3^k/(1-x^3^k). - Ralf Stephan, Apr 12 2002
Fixed point of the morphism: 1 -> 112; 2 -> 113; 3 -> 114; 4 -> 115; ...; starting from a(1) = 1. a(3n+1) = a(3n+2) = 1; a(3n) = 1 + a(n). - Philippe Deléham, Mar 29 2004
a(n) = (-1)*Sum_{d divides n} mu(3d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/(1-1/3^s). - R. J. Mathar, Jun 13 2011
a(n) = (1/2)*(3 - A053735(n) + A053735(n-1)) for n >= 1. - Tom Edgar, Aug 06 2014
a(n) = A007949(3n). - Cyril Damamme, Aug 04 2015
a(2n) = a(n), a(2n-1) = A254046(n). - Cyril Damamme, Aug 04 2015
G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x). - Ilya Gutkovskiy, May 03 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Sep 11 2020 [corrected by Vaclav Kotesovec, Jun 25 2024, see also A004128]
a(n) = tau(n)/(tau(3*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
G.f.: Sum_{i>=1, j>=0} x^(i*3^j). - Seiichi Manyama, Mar 23 2025
Conjecture: a(n) = A007949(A000045(4*n)), all other 3-adic quadrisections A007949(A000045(.))=0. [Lengyel?]. - R. J. Mathar, Jun 28 2025

More terms from James Sellers, Dec 11 1999

More terms from Vladeta Jovovic, Aug 24 2002

A006047 Number of entries in n-th row of Pascal's triangle not divisible by 3.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 3, 6, 9, 2, 4, 6, 4, 8, 12, 6, 12, 18, 3, 6, 9, 6, 12, 18, 9, 18, 27, 2, 4, 6, 4, 8, 12, 6, 12, 18, 4, 8, 12, 8, 16, 24, 12, 24, 36, 6, 12, 18, 12, 24, 36, 18, 36, 54, 3, 6, 9, 6, 12, 18, 9, 18, 27, 6, 12, 18, 12, 24, 36, 18, 36, 54, 9, 18, 27, 18, 36, 54, 27, 54

Offset: 0

Jeffrey Shallit

Fixed point of the morphism a -> a, 2a, 3a, starting from a(1) = 1. - Robert G. Wilson v, Jan 24 2006
This is a particular case of the number of entries in n-th row of Pascal's triangle not divisible by a prime p, which is given by a simple recursion using ⊗, the Kronecker (or tensor) product of vectors. Let v_0=(1,2,...,p). Then v_{n+1}=v_0 ⊗ v_n, where the vector v_n contains the values for the first p^n rows of Pascal's triangle (rows 0 through p^n-1). - William B. Everett (bill(AT)chgnet.ru), Mar 29 2008
a(n) = A206424(n) + A227428(n); number of nonzero terms in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013

			15 in base 3 is 120, here r=1 and s=1 so a(15) = 3*2 = 6.
William B. Everett's comment with p=3, n=2: v_0 = (1,2,3), v_1 = (1,2,3) => v_2 = (1*1,1*2,1*3,2*1,2*2,2*3,3*1,3*2,3*3) = (1,2,3,2,4,6,3,6,9), the first 3^2 values of the present sequence. - _Wolfdieter Lang_, Mar 19 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 0..19683 (terms 0..1000 from Reinhard Zumkeller).
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Michael Gilleland, Some Self-Similar Integer Sequences
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62.1 (1977), 19-22. (Annotated scanned copy)
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024.
Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
Index entries for sequences that are fixed points of mappings

Cf. A001316, A003586, A038148, A053735, A083093, A089898, A206424, A227428, A286586, A286587, A286633.

a006047 = sum . map signum . a083093_row
-- Reinhard Zumkeller, Jul 11 2013

p:=proc(n) local ct, k: ct:=0: for k from 0 to n do if binomial(n,k) mod 3 = 0 then else ct:=ct+1 fi od: end: seq(p(n),n=0..82); # Emeric Deutsch
f:= proc(n) option remember; ((n mod 3)+1)*procname(ceil((n+1)/3)-1) end proc:
f(0):= 1: f(1):= 2:
seq(f(i), i=0..100); # Robert Israel, Oct 15 2015

Nest[Flatten[ # /. a_Integer -> {a, 2a, 3a}] &, {1}, 4] (* Robert G. Wilson v, Jan 24 2006 *)
Nest[ Join[#, 2#, 3#] &, {1}, 4] (* Robert G. Wilson v, Jul 27 2014 *)

b(n)=if(n<3,n,if(n%3==0,3*b(n/3),if(n%3==1,1*b((n+2)/3),2*b((n+1)/3)))) \\ Ralf Stephan

A006047(n) = b(1+n); \\ (The above PARI-program by Ralf Stephan is for offset-1-version of this sequence.) - Antti Karttunen, May 28 2017

A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n \= 3); m; }; \\ Antti Karttunen, May 28 2017

a(n) = prod(i=1,#d=digits(n, 3), (1+d[i])) \\ David A. Corneth, May 28 2017

upto(n) = my(res = [1], v); while(#res < n, v = concat(2*res, 3*res); res = concat(res, v)); res \\ David A. Corneth, May 29 2017

from sympy.ntheory.factor_ import digits
from sympy import prod
def a(n):
    d=digits(n, 3)
    return n + 1 if n<3 else prod(1 + d[i] for i in range(1, len(d)))
print([a(n) for n in range(51)]) # Indranil Ghosh, Jun 06 2017

from sympy.ntheory import digits
def A006047(n): return 3**(s:=digits(n,3)).count(2)<Chai Wah Wu, Apr 24 2025

(define (A006047 n) (if (zero? n) 1 (let ((d (mod n 3))) (* (+ 1 d) (A006047 (/ (- n d) 3)))))) ;; For R6RS standard. Use modulo instead of mod in older Schemes like MIT/GNU Scheme. - Antti Karttunen, May 28 2017

Write n in base 3; if the representation contains r 1's and s 2's then a(n) = 3^s * 2^r. Also a(n) = Sum_{k=0..n} (C(n, k)^2 mod 3). - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
a(n) = b(n+1), with b(1)=1, b(2)=2, b(3n)=3b(n), b(3n+1)=b(n+1), b(3n+2)=2b(n+1). - Ralf Stephan, Sep 15 2003
G.f.: Product_{n>=0} (1+2*x^(3^n)+3*x^(2*3^n)) (Northshield). - Johannes W. Meijer, Jun 05 2011
G.f. g(x) satisfies g(x) = (1 + 2*x + 3*x^2)*g(x^3). - Robert Israel, Oct 15 2015
From Tom Edgar, Oct 15 2015: (Start)
a(3^k) = 2 for k>=0;
a(2*3^k) = 3 for k>=0;
a(n) = Product_{b_j != 0} a(b_j*3^j) where n = Sum_{j>=0} b_j*3^j is the ternary representation of n. (End)
A056239(a(n)) = A053735(n). - Antti Karttunen, Jun 03 2017
a(n) = Sum_{k = 0..n} mod(C(n,k)^2, 3). - Peter Bala, Dec 17 2020

More terms from Ralf Stephan, Sep 15 2003

A053824 Sum of digits of (n written in base 5).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 4, 5, 6

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4}, 1->{1,2,3,4,5}, 2->{2,3,4,5,6}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 4 + 0 = 4 because 20 is written as 40 in base 5.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle:
  0,
  1,2,3,4,
  1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,
  1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9,2,3,4,5,6,3,4,5,6,7,4,5,...
See the conjecture in the entry A000120. (End)

Tanar Ulric, Table of n, a(n) for n = 0..10000 (terms 0..3125=5^5 from Reinhard Zumkeller).
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.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A231668, A231669, A231670, A231671, A138530.
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, this sequence, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173525. - Omar E. Pol, Feb 21 2010
Cf. A173670 (last nonzero decimal digit of (10^n)!). - Washington Bomfim, Jan 01 2011

a053824 0 = 0
a053824 x = a053824 x' + d  where (x', d) = divMod x 5
-- Reinhard Zumkeller, Jan 31 2014

[&+Intseq(n, 5):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* or *)
Nest[Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 4}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
f[n_] := n - 4 Sum[Floor[n/5^k], {k, n}]; Array[f, 103, 0]

PARI
```
a(n)=if(n<1,0,if(n%5,a(n-1)+1,a(n/5)))
```

a(n) = sumdigits(n, 5); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(5n+i) = a(n) + i for 0 <= i <= 4;
a(n) = n - 4*Sum_{k>=1} floor(n/5^k) = n - 4*A027868(n). (End)
a(n) = A138530(n,5) for n > 4. - Reinhard Zumkeller, Mar 26 2008
If i >= 2, a(2^i) mod 4 = 0. - Washington Bomfim, Jan 01 2011
a(n) = Sum_{k>=0} A031235(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 5^floor(log_5(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 5*log(5)/4 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A003137 Write n in base 3 and juxtapose.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

An irregular table in which the n-th row lists the base-3 digits of n, see A007089. - Jason Kimberley, Dec 07 2012

The base-3 Champernowne constant (A077771): it is normal in base 3. - Jason Kimberley, Dec 07 2012

			1,
2,
1,0,
1,1,
1,2,
2,0,
2,1,
2,2,
1,0,0,
1,0,1,.... _R. J. Mathar_, Aug 16 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), this sequence and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A081604 (row lengths), A053735 (row sums), A030341 (rows reversed), A077771, A007089.

a003137 n k = a003137_tabf !! (n-1) !! k
a003137_row n = a003137_tabf !! (n-1)
a003137_tabf = map reverse $ tail a030341_tabf
a003137_list = concat a003137_tabf
-- Reinhard Zumkeller, Feb 21 2013

&cat[Reverse(IntegerToSequence(n,3)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 40, 3] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ a[#, 3] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A003137_gen(): return (d for m in count(1) for d in digits(m,3)[1:])
A003137_list = list(islice(A003137_gen(),30)) # Chai Wah Wu, Jan 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A065363 Sum of balanced ternary digits in n. Replace 3^k with 1 in balanced ternary expansion of n.

Original entry on oeis.org

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

Offset: 0

Marc LeBrun, Oct 31 2001

Notation: (3)(1).
Extension to negative n: a(-n) = -a(n). - Franklin T. Adams-Watters, May 13 2009
Row sums of A059095. - Rémy Sigrist, Oct 05 2019

			5 = + 1(9) - 1(3) - 1(1) -> +1 - 1 - 1 = -1 = a(5).

Reinhard Zumkeller, 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 12 (2009) 09.5.6.
Daniel Forgues, Table of n, a(n) for n = 0..100000

Cf. A059095, A065364, A053735. See A134452 for iterations.

Cf. A134022, A134024.

a:= proc(n) `if`(n=0, 0, (d-> `if`(d=2,
      a(q+1)-1, d+a(q)))(irem(n, 3, 'q')))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Jan 09 2020

balTernDigits[0] := {0}; balTernDigits[n_/;n > 0] := Module[{unParsed = n, currRem, currExp = 1, digitList = {}, nextDigit}, While[unParsed > 0, If[unParsed == 3^(currExp - 1), digitList = Append[digitList, 1]; unParsed = 0, currRem = Mod[unParsed, 3^currExp]/3^(currExp - 1); nextDigit = Switch[currRem, 0, 0, 2, -1, 1, 1]; digitList = Append[digitList, nextDigit]; unParsed = unParsed - nextDigit * 3^(currExp - 1)]; currExp++]; digitList = Reverse[digitList]; Return[digitList]]; balTernDigits[n_/;n < 0] := (-1)balTernDigits[Abs[n]]; Table[Plus@@balTernDigits[n], {n, 0, 108}] (* Alonso del Arte, Feb 25 2011 *)
terVal[lst_List] := Reverse[lst].(3^Range[0, Length[lst] - 1]); maxDig = 4; t = Table[0, {3 * 3^maxDig/2}]; t[[1]] = 1; Do[d = IntegerDigits[Range[0, 3^dig - 1], 3, dig]/.{2 -> -1}; d = Prepend[#, 1]&/@d; t[[terVal/@d]] = Total/@d, {dig, maxDig}]; Prepend[t, 0] (* T. D. Noe, Feb 24 2011 *)
Array[Total[Prepend[IntegerDigits[#, 3], 0] //. {a___, b_, 2, c___} :> {a, b + 1, -1, c}] &, 109, 0] (* Michael De Vlieger, Jun 27 2020 *)

bt(n)=my(d=digits(n,3),c=1); while(c, if(d[1]==2, d=concat(0,d)); c=0; for(i=2,#d, if(d[i]==2,d[i]=-1;d[i-1]+=1;c=1))); d
a(n)=vecsum(bt(n)) \\ Charles R Greathouse IV, May 07 2020

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

G.f.: (1/(1-x))*Sum_{k>=0} (x^3^k - x^(2*3^k))/(x^((3^k-1)/2)*(1 + x^3^k + x^(2*3^k))). - Franklin T. Adams-Watters, May 13 2009
a(n) = A134024(n) - A134022(n). - Reinhard Zumkeller, Dec 16 2010
a(3*n - 1) = a(n) - 1, a(3*n) = a(n), a(3*n + 1) = a(n) + 1. - Thomas König, Jun 24 2020
a(n) = A053735(2n) - A053735(n). This can be shown with constructing balanced ternary representation of n: Add n and n with carry, and then subtract n from the sum without borrow. - Yifan Xie, Dec 24 2024

A053830 Sum of digits of (n written in base 9).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8}, 1->{1,2,3,4,5,6,7,8,9}, 2->{2,3,4,5,6,7,8,9,10}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 2+2 = 4 because 20 is written as 22 base 9.
From _Omar E. Pol_, Feb 23 2010: (Start)
It appears that this can be written as a triangle (see the conjecture in the entry A000120):
0;
1,2,3,4,5,6,7,8;
1,2,3,4,5,6,7,8,9,2,3,4,5,6,7,8,9,10,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,...
where the rows converge to A173529. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..59049
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.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A053735, A053737, A053824, A053828, A231684, A231685, A231686, A231687.

Cf. A173529. - Omar E. Pol, Feb 23 2010

[&+Intseq(n, 9):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 9], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 8}]] &, {0}, 3] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%9,a(n-1)+1,a(n/9)))
```

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(9n+i) = a(n) + i for 0 <= i <= 8;
a(n) = n - 8*Sum_{k>=1} floor(n/9^k) = n - 8*A054898(n). (End)
a(n) = A138530(n,9) for n > 8. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A031087(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 9^floor(log_9(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 9*log(9)/8 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A053827 Sum of digits of (n written in base 6).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5}, 1->{1,2,3,4,5,6}, 2->{2,3,4,5,6,7}, etc. - Robert G. Wilson v, Jul 27 2006

Sum of six consecutive terms is (15,21,27,33,39,45; 21,27,33,39,45,51; 27,33,39,45,51,57; and so on). - Vincenzo Librandi, Aug 02 2010

			a(20)=3+2=5 because 20 is written as 32 base 6.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle :
  0,
  1,2,3,4,5,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,
  1,2,3,4,5,6,2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11,2...
where the rows converge to A173526.
See the conjecture in the entry A000120. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..10000
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.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A231672, A231673, A231674, A231675, A138530.
Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, this sequence, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Cf. A173526. - Omar E. Pol, Feb 21 2010

[&+Intseq(n,6):n in [0..105]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 6], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 5}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%6,a(n-1)+1,a(n/6)))
```

a(n) = sumdigits(n, 6); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(6n+i) = a(n)+i for 0 <= i <= 5.
a(n) = n-5*(Sum_{k>0} floor(n/6^k)) = n-5*A054895(n). (End)
a(n) = A138530(n,6) for n > 5. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030567(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 6^floor(log_6(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 6*log(6)/5 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 30, 31, 32, 34, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 72, 74, 75, 76, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 0

N. J. A. Sloane

nonn

3-adic valuation of (3n)!; cf. A054861.

Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are in |A067622|.

Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth and Number", by Jay Kappraff, World Scientific, 2002, p. 356.

T. D. Noe, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004117, A001511, A051064, A055457.

Cf. A054861, A067080, A098844, A132027, A005187, A054899.

a004128 n = a004128_list !! (n-1)
a004128_list = scanl (+) 0 a051064_list
-- Reinhard Zumkeller, May 23 2013

[n + Valuation(Factorial(n), 3): n in [0..70]]; // Vincenzo Librandi, Jun 12 2019

A004128 := proc(n)
    A054861(3*n) ;
end proc:
seq(A004128(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Table[Total[NestWhileList[Floor[#/3] &, n, # > 0 &]], {n, 0, 70}] (* Birkas Gyorgy, May 20 2012 *)
A004128 = Log[3, CoefficientList[ Series[1/(1+x)^(1/3), {x, 0, 100}], x] // Denominator] (* Jean-François Alcover, Feb 19 2015 *)
Flatten[{0, Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]]}] (* Vaclav Kotesovec, Oct 17 2019 *)

{a(n) = my(s, t=1); while(t<=n, s += n\t; t*=3);s}; /* Michael Somos, Feb 26 2004 */

a(n) = (3*n-sumdigits(n,3))/2; \\ Christian Krause, Jun 10 2025

def A007949(n):
    c = 0
    while not (a:=divmod(n,3))[1]:
        c += 1
        n = a[0]
    return c
def A004128(n): return n+sum(A007949(i) for i in range(3,n+1)) # Chai Wah Wu, Feb 28 2025

A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
[A004128(n) for n in (0..70)]  # Peter Luschny, Nov 16 2012

A051064(n) = a(n+1) - a(n). - Alford Arnold, Jul 19 2000
a(n) = n + floor(n/3) + floor(n/9) + floor(n/27) + ... = n + a(floor(n/3)) = n + A054861(n) = A054861(3n) = (3*n - A053735(n))/2. - Henry Bottomley, May 01 2001
a(n) = Sum_{k>=0} floor(n/3^k). a(n) = Sum_{k=0..floor(log_3(n))} floor(n/3^k), n >= 1. - Hieronymus Fischer, Aug 14 2007
Recurrence: a(n) = n + a(floor(n/3)); a(3n) = 3*n + a(n); a(n*3^m) = 3*n*(3^m-1)/2 + a(n). - Hieronymus Fischer, Aug 14 2007
a(k*3^m) = k*(3^(m+1)-1)/2, 0 <= k < 3, m >= 0. - Hieronymus Fischer, Aug 14 2007
Asymptotic behavior: a(n) = (3/2)*n + O(log(n)), a(n+1) - a(n) = O(log(n)); this follows from the inequalities below. - Hieronymus Fischer, Aug 14 2007
a(n) <= (3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer, Aug 14 2007
a(n) >= (3n-2)/2 - floor(log_3(n)); equality holds for n = 3^m - 1, m > 0. - Hieronymus Fischer, Aug 14 2007
Lim inf (3n/2 - a(n)) = 1/2, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (3n/2 - log_3(n) - a(n)) = 0, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (a(n+1) - a(n) - log_3(n)) = 1, for n->oo. - Hieronymus Fischer, Aug 14 2007
G.f.: (Sum_{k>=0} x^(3^k)/(1-x^(3^k)))/(1-x). - Hieronymus Fischer, Aug 14 2007
a(n) = Sum_{k>=0} A030341(n,k)*A003462(k+1). - Philippe Deléham, Oct 21 2011
a(n) ~ 3*n/2 - log(n)/(2*log(3)). - Vaclav Kotesovec, Oct 17 2019

Current definition suggested by Jason Earls, Jul 04 2001

cp's OEIS Frontend

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

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A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023.

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A051064 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.

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A006047 Number of entries in n-th row of Pascal's triangle not divisible by 3.

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A053824 Sum of digits of (n written in base 5).

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