A006047 -id:A006047 - cp's OEIS frontend

A083093 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 3.

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

Offset: 0

Benoit Cloitre, Apr 22 2003

Start with [1], repeatedly apply the map 0 -> [000/000/000], 1 -> [111/120/100], 2 -> [222/210/200]. - Philippe Deléham, Apr 16 2009

{T(n,k)} is a fractal gasket with fractal (Hausdorff) dimension log(A000217(3))/log(3) = log(6)/log(3) = 1.63092... (see Reiter reference). Replacing values greater than 1 with 1 produces a binary gasket with the same dimension (see Bondarenko reference). - Richard L. Ollerton, Dec 14 2021

			.            Rows 0 .. 3^3:
.    0:                             1
.    1:                            1 1
.    2:                           1 2 1
.    3:                          1 0 0 1
.    4:                         1 1 0 1 1
.    5:                        1 2 1 1 2 1
.    6:                       1 0 0 2 0 0 1
.    7:                      1 1 0 2 2 0 1 1
.    8:                     1 2 1 2 1 2 1 2 1
.    9:                    1 0 0 0 0 0 0 0 0 1
.   10:                   1 1 0 0 0 0 0 0 0 1 1
.   11:                  1 2 1 0 0 0 0 0 0 1 2 1
.   12:                 1 0 0 1 0 0 0 0 0 1 0 0 1
.   13:                1 1 0 1 1 0 0 0 0 1 1 0 1 1
.   14:               1 2 1 1 2 1 0 0 0 1 2 1 1 2 1
.   15:              1 0 0 2 0 0 1 0 0 1 0 0 2 0 0 1
.   16:             1 1 0 2 2 0 1 1 0 1 1 0 2 2 0 1 1
.   17:            1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1
.   18:           1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1
.   19:          1 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 1 1
.   20:         1 2 1 0 0 0 0 0 0 2 1 2 0 0 0 0 0 0 1 2 1
.   21:        1 0 0 1 0 0 0 0 0 2 0 0 2 0 0 0 0 0 1 0 0 1
.   22:       1 1 0 1 1 0 0 0 0 2 2 0 2 2 0 0 0 0 1 1 0 1 1
.   23:      1 2 1 1 2 1 0 0 0 2 1 2 2 1 2 0 0 0 1 2 1 1 2 1
.   24:     1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1
.   25:    1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1
.   26:   1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
.   27:  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 .
- _Reinhard Zumkeller_, Jul 11 2013

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J.-P. Allouche, F. von Haeseler, H.-O. Peitgen, and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Disc. Appl. Math. 66 (1996) 1-22.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
Lin Jiu and Christophe Vignat, On Binomial Identities in Arbitrary Bases, arXiv:1602.04149 [math.CO], 2016.
Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A051638 (row sums), A090044, A047999, A034931, A034930, A008975, A034932, A062296, A006047.
Cf. A006996 (central terms), A173019, A206424, A227428.
Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), (this sequence) (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

a083093 n k = a083093_tabl !! n !! k
a083093_row n = a083093_tabl !! n
a083093_tabl = iterate
   (\ws -> zipWith (\u v -> mod (u + v) 3) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Jul 11 2013

/* As triangle: */ [[Binomial(n,k) mod 3: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

A083093 := proc(n,k)
    modp(binomial(n,k),3) ;
end proc:
seq(seq(A083093(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jul 26 2017

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 3] (* Robert G. Wilson v, Jan 19 2004 *)

from sympy import binomial
def T(n, k):
    return binomial(n, k) % 3
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jul 26 2017

from math import comb, isqrt
def A083093(n):
    def f(m,k):
        if m<3 and k<3: return comb(m,k)%3
        c,a = divmod(m,3)
        d,b = divmod(k,3)
        return f(c,d)*f(a,b)%3
    return f(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2)) # Chai Wah Wu, Apr 30 2025

T(i, j) = binomial(i, j) mod 3.
T(n+1,k) = (T(n,k) + T(n,k-1)) mod 3. - Reinhard Zumkeller, Jul 11 2013
T(n,k) = Product_{i>=0} binomial(n_i,k_i) mod 3, where n = Sum_{i>=0} n_i*3^i and k = Sum_{i>=0} k_i*3^i, 0<=n_i, k_i <=2 [Allouche et al.]. - R. J. Mathar, Jul 26 2017

A194459 Number of entries in the n-th row of Pascal's triangle not divisible by 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 6, 12, 18, 24, 30, 8, 16, 24, 32, 40, 10, 20, 30, 40, 50, 3, 6, 9, 12, 15, 6, 12, 18, 24, 30, 9, 18, 27, 36, 45, 12, 24, 36, 48, 60, 15, 30

Offset: 0

Paul Weisenhorn, Aug 24 2011

Pascal triangles modulo p with p prime have the dimension D = log(p*(p+1)/2)/log(p). [Corrected by Connor Lane, Nov 28 2022]

Also number of ones in row n of triangle A254609. - Reinhard Zumkeller, Feb 04 2015

			n = 32 = 112|_5: b(32,1) = 2, b(32,2) = 1, thus a(32) = 2^2 * 3^1 = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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.

Cf. A006046, A001316 (for p=2).
Cf. A006048, A006047 (for p=3).
Cf. A194458 (for p=5).
Cf. A007318, A254609.

a194459 = sum . map (signum . flip mod 5) . a007318_row
-- Reinhard Zumkeller, Feb 04 2015

a:= proc(n) local l, m, t;
      m:= n;
      l:= [0$5];
      while m>0 do t:= irem(m, 5, 'm')+1; l[t]:=l[t]+1 od;
      mul(r^l[r], r=2..5)
    end:
seq(a(n), n=0..100);

Nest[Join[#, 2#, 3#, 4#, 5#]&, {1}, 4] (* Jean-François Alcover, Apr 12 2017, after code by Robert G. Wilson v in A006047 *)

from math import prod
from sympy.ntheory import digits
def A194459(n):
    s = digits(n,5)[1:]
    return prod((d+1)**s.count(d) for d in range(1,5)) # Chai Wah Wu, Jul 23 2025

a(n) = Product_{d=1..4} (d+1)^b(n,d) with b(n,d) = number of digits d in base 5 expansion of n. The formula generalizes to other prime bases p.

a(n) = A194458(n) - A194458(n-1).

Edited by Alois P. Heinz, Sep 06 2011

A382720 Number of entries in the n-th row of Pascal's triangle not divisible by 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 4, 6, 8, 10, 12, 14, 3, 6, 9, 12, 15, 18, 21, 4, 8, 12, 16, 20, 24, 28, 5, 10, 15, 20, 25, 30, 35, 6, 12, 18, 24, 30, 36, 42, 7, 14, 21, 28, 35, 42, 49, 2, 4, 6, 8, 10, 12, 14, 4, 8, 12, 16, 20, 24, 28, 6, 12, 18, 24, 30, 36, 42, 8, 16, 24, 32, 40, 48, 56, 10, 20, 30, 40

Offset: 0

N. J. A. Sloane, Apr 23 2025

nonn

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.

Cf. A001316, A006047, A194459, A382721-A382722, A382726.

a[n_] := Times @@ (IntegerDigits[n, 7] + 1);Array[a,81,0] (* James C. McMahon, Aug 16 2025 *)

from math import prod
from gmpy2 import digits
def A382720(n): return prod(int(d)+1 for d in digits(n,7)) # Chai Wah Wu, Aug 10 2025

A051638 a(n) = Sum_{k=0..n} (C(n,k) mod 3).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 4, 8, 13, 2, 4, 8, 4, 8, 16, 8, 16, 26, 4, 8, 13, 8, 16, 26, 13, 26, 40, 2, 4, 8, 4, 8, 16, 8, 16, 26, 4, 8, 16, 8, 16, 32, 16, 32, 52, 8, 16, 26, 16, 32, 52, 26, 52, 80, 4, 8, 13, 8, 16, 26, 13, 26, 40, 8, 16, 26, 16, 32, 52, 26, 52, 80, 13

Offset: 0

N. J. A. Sloane

Row sums of the triangle in A083093. - Reinhard Zumkeller, Jul 11 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..6561=3^8
Michael Gilleland, Some Self-Similar Integer Sequences
A. Granville, Binomials modulo a prime

Cf. A001316.

a051638 = sum . a083093_row  -- Reinhard Zumkeller, Jul 11 2013

Table[2^(DigitCount[n,3,1]-1) (3^(DigitCount[n,3,2]+1)-1),{n,0,80}] (* Harvey P. Dale, Jun 20 2019 *)

from gmpy2 import digits
def A051638(n): return 3*3**(s:=digits(n,3)).count('2')-1<>1 # Chai Wah Wu, Jun 25 2025

Write n in base 3; if the representation contains r 1's and s 2's then a(n) = 2^{r-1} * (3^(s+1) - 1) = 1/2 * (3*A006047(n) - 2^(A062756(n))). - Robin Chapman, Ahmed Fares (ahmedfares(AT)my-deja.com) and others, Jul 16 2001

a(3n) = a(n), a(3n+1) = 2a(n), a(9n+2) = a(3n+2), a(9n+5) = 2a(3n+2), a(9n+8) = 4a(3n+2) - 3a(n). - David Radcliffe, Jun 25 2025

A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 3, 10, 5, 2, 5, 7, 3, 21, 5, 10, 36, 14, 5, 27, 12, 2, 5, 16, 5, 14, 23, 7, 12, 29, 3, 21, 5, 21, 78, 27, 5, 27, 12, 10, 78, 14, 36, 136, 44, 14, 90, 25, 5, 27, 23, 27, 90, 61, 12, 42, 38, 2, 5, 16, 5, 14, 23, 16, 23, 67, 5, 27, 23, 14, 44, 40, 23, 61, 80, 7, 12, 67, 12, 25, 80, 29, 38, 121, 3, 21, 5, 21, 78, 27, 5, 27, 12, 21, 465, 27, 78, 300, 90, 27

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

For all i, j: a(i) = a(j) => A006047(i) = A006047(j) => A053735(i) = A053735(j).

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

Cf. A000027, A278222, A289813, A289814, A290091, A290092.

Cf. A006047, A053735, A290079 (some of the matched sequences).

(define (A290093 n) (* (/ 1 2) (+ (expt (+ (A290091 n) (A290092 n)) 2) (- (A290091 n)) (- (* 3 (A290092 n))) 2)))

a(n) = (1/2)*(2 + ((A290091(n)+A290092(n))^2) - A290091(n) - 3*A290092(n)).

A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3.

Original entry on oeis.org

1, 3, 6, 8, 12, 18, 21, 27, 36, 38, 42, 48, 52, 60, 72, 78, 90, 108, 111, 117, 126, 132, 144, 162, 171, 189, 216, 218, 222, 228, 232, 240, 252, 258, 270, 288, 292, 300, 312, 320, 336, 360, 372, 396, 432, 438, 450, 468, 480, 504, 540, 558, 594, 648, 651, 657, 666, 672, 684, 702, 711, 729, 756, 762, 774, 792, 804, 828, 864, 882, 918, 972, 981, 999, 1026, 1044, 1080, 1134, 1161, 1215, 1296

Offset: 0

Jeffrey Shallit

nonn

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

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
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. See p. 1.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & graphics 13.1 (1989): 59-62.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-62. (Annotated scanned copy)
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-60. (Annotated scanned copy)
A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.

Partial sums of A006047.

from math import prod
from gmpy2 import digits
def A006048(n): return sum(prod(int(d)+1 for d in digits(m,3)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A006048(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,3)[::-1]))
    return sum((b-1)*prod(d[a:])*6**a for a, b in enumerate(d))>>1  # Chai Wah Wu, Aug 13 2025

More terms from N. J. A. Sloane, Apr 23 2025

A062296 a(n) = number of entries in n-th row of Pascal's triangle divisible by 3.

Original entry on oeis.org

0, 0, 0, 2, 1, 0, 4, 2, 0, 8, 7, 6, 9, 6, 3, 10, 5, 0, 16, 14, 12, 16, 11, 6, 16, 8, 0, 26, 25, 24, 27, 24, 21, 28, 23, 18, 33, 30, 27, 32, 25, 18, 31, 20, 9, 40, 35, 30, 37, 26, 15, 34, 17, 0, 52, 50, 48, 52, 47, 42, 52, 44, 36, 58, 53, 48, 55, 44, 33, 52, 35, 18, 64, 56, 48, 58, 41

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

Number of zeros in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013

			When n=3 the row is 1,3,3,1 so a(3) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
D. L. Wells, Residue counts modulo three for the fibonacci triangle, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.

Cf. A006047, A083093, A206424, A227428.

Cf. A062756, A081603.

a062296 = sum . map ((1 -) . 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(n+1-p(n),n=0..83); # Emeric Deutsch

a[n_] := Count[(Binomial[n, #] & )[Range[0, n]], _?(Divisible[#, 3] & )];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 26 2018 *)
Table[n + 1 - 2^(DigitCount[n, 3, 1])*3^(DigitCount[n, 3, 2]), {n, 0, 76}] (* Shenghui Yang, Jan 08 2025 *)

from sympy.ntheory import digits
def A062296(n):
    s = digits(n,3)[1:]
    return n+1-(3**s.count(2)<Chai Wah Wu, Jul 24 2025

a(n) = n + 1 - A006047(n).
a(n) = n + 1 - A206424(n) - A227428(n). - Reinhard Zumkeller, Jul 11 2013
a(n) = n + 1 - 2^A062756(n)*3^A081603(n). - Shenghui Yang, Jan 08 2025

More terms from Emeric Deutsch, Feb 03 2005

A194458 Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.

Original entry on oeis.org

1, 3, 6, 10, 15, 17, 21, 27, 35, 45, 48, 54, 63, 75, 90, 94, 102, 114, 130, 150, 155, 165, 180, 200, 225, 227, 231, 237, 245, 255, 259, 267, 279, 295, 315, 321, 333, 351, 375, 405, 413, 429, 453, 485, 525, 535, 555, 585, 625, 675, 678, 684, 693, 705, 720, 726

Offset: 0

Paul Weisenhorn, Aug 24 2011

nonn

The number of zeros in the first n rows is binomial(n+1,2) - a(n).

			n = 38: n+1 = 39 = 124_5, thus a(38) = (C(5,2)*15^0*3 + C(3,2)*15^1)*2 + C(2,2)*15^2 = (10*1*3 + 3*15)*2 + 1*225 = 375.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 53.
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. See p. 1.

A006046(n+1) = A006046(n) + A001316(n) for p=2.
A006048(n+1) = A006048(n) + A006047(n+1) for p=3.
a(n+1) = a(n) + A194459(n+1) for p=5.

a:= proc(n) local l, m, h, j;
      m:= n+1;
      l:= [];
      while m>0 do l:= [l[], irem (m, 5, 'm')+1] od;
      h:= 0;
      for j to nops(l) do h:= h*l[j] +binomial (l[j], 2) *15^(j-1) od:
      h
    end:
seq(a(n), n=0..100);

a[n_] := Module[{l, m, r, h, j}, m = n+1; l = {}; While[m>0, l = Append[l, {m, r} = QuotientRemainder[m, 5]; r+1]]; h = 0; For[j = 1, j <= Length[l], j++, h = h*l[[j]] + Binomial [l[[j]], 2] *15^(j-1)]; h]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)

from math import prod
from gmpy2 import digits
def A194458(n): return sum(prod(int(d)+1 for d in digits(m,5)) for m in range(n+1)) # Chai Wah Wu, Aug 10 2025

from math import prod
from gmpy2 import digits
def A194458(n):
    d = list(map(lambda x:int(x)+1,digits(n+1,5)[::-1]))
    return sum((b-1)*prod(d[a:])*15**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

a(n) = ((C(d0+1,2)*15^0*(d1+1) + C(d1+1,2)*15^1)*(d1+1) + C(d1+1,2)*15^1)*(d2+1) + C(d2+1,2)*15^2 ..., where d_k...d_1d_0 is the base 5 expansion of n+1 and 15 = binomial(5+1,2). The formula generalizes to other prime bases p.

Edited by Alois P. Heinz, Sep 06 2011

A038148 Number of 3-infinitary divisors of n: if n = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-infinitary-divisor of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 4, 8

Offset: 1

Yasutoshi Kohmoto

Multiplicative: If e = sum d_k 3^k, then a(p^e) = prod (d_k+1). - Christian G. Bower, May 19 2005

			2^3*3 is a 3-infinitary-divisor of 2^5*3^2 because 2^3*3 = 2^10*3^1 and 2^5*3^2 = 2^12*3^2 in ternary expanded power. All corresponding digits satisfy the condition. 1 <= 1, 0 <= 2, 1 <= 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Index entries for sequences computed from exponents in factorization of n

Cf. A006047, A037445, A038182, A074848.

A038148 := proc(n) if n= 1 then 1; else ifa := ifactors(n)[2] ;
a := 1; for f in ifa do e := convert(op(2,f),base,3) ; a := a*mul(d+1,d=e) ; end do: end if; end proc:
seq(A038148(n),n=1..50) ; # R. J. Mathar, Feb 08 2011

a[1] = 1; a[n_] := (k = 1; Do[k = k * Times @@ (IntegerDigits[f, 3] + 1), {f, FactorInteger[n][[All, 2]]}]; k); Table[a[n], {n, 1, 102}](* Jean-François Alcover, Feb 03 2012, after R. J. Mathar *)

A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n = (n-d)/3); m; };
A038148(n) = factorback(apply(e -> A006047(e), factorint(n)[, 2])); \\ (After A037445) - Antti Karttunen, May 28 2017

(define (A038148 n) (if (= 1 n) n (* (A006047 (A067029 n)) (A038148 (A028234 n))))) ;; Antti Karttunen, May 28 2017

a(1) = 1; for n > 1, a(n) = A006047(A067029(n)) * a(A028234(n)). [After Christian G. Bower's 2005 comment.] - Antti Karttunen, May 28 2017

More terms from Naohiro Nomoto, Jun 21 2001

Data section further extended to 105 terms by Antti Karttunen, May 28 2017

A206424 The number of 1's in row n of Pascal's Triangle (mod 3).

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 4, 5, 2, 4, 4, 4, 8, 8, 4, 8, 10, 2, 4, 5, 4, 8, 10, 5, 10, 14, 2, 4, 4, 4, 8, 8, 4, 8, 10, 4, 8, 8, 8, 16, 16, 8, 16, 20, 4, 8, 10, 8, 16, 20, 10, 20, 28, 2, 4, 5, 4, 8, 10, 5, 10, 14, 4, 8, 10, 8, 16, 20, 10, 20, 28, 5, 10, 14, 10, 20, 28

Offset: 0

Marcus Jaiclin, Feb 07 2012

A006047(n) = a(n) + A227428(n).

a(n) = n + 1 - A062296(n) - A227428(n); number of ones in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013

			Rows 0-8 of Pascal's Triangle (mod 3) are:
  1                   So a(0) = 1
  1 1                 So a(1) = 2
  1 2 1               So a(2) = 2
  1 0 0 1                 .
  1 1 0 1 1               .
  1 2 1 1 2 1             .
  1 0 0 2 0 0 1
  1 1 0 2 2 0 1 1
  1 2 1 2 1 2 1 2 1

Reinhard Zumkeller (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..19683
R. Garfield and H. S. Wilf, The distribution of the binomial coefficients modulo p, J. Numb. Theory 41 (1) (1992) 1-5
Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5
D. L. Wells, Residue counts modulo three for the fibonacci triangle, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.
Avery Wilson, Pascal's Triangle Modulo 3, Mathematics Spectrum, 47-2 - January 2015, pp. 72-75.

Cf. A083093, A062296, A006047, A062756, A081603, A206427, A227428.

a206424 = length . filter (== 1) . a083093_row
-- Reinhard Zumkeller, Jul 11 2013

Table[Count[Mod[Binomial[n, Range[0, n]], 3], 1], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)

A206424(n) = sum(k=0,n,1==(binomial(n,k)%3)); \\ (naive way) Antti Karttunen, Jul 26 2017

from sympy.ntheory import digits
def A206424(n):
    s = digits(n,3)[1:]
    return (3**s.count(2)+1)<>1 # Chai Wah Wu, Jul 24 2025

(define (A206424 n) (* (+ (A000244 (A081603 n)) 1) (A000079 (- (A062756 n) 1)))) ;; (fast way) Antti Karttunen, Jul 27 2017

From Antti Karttunen, Jul 27 2017: (Start)
a(n) = (3^k + 1)*2^(y-1), where y = A062756(n) and k = A081603(n). [See e.g. Wells or Wilson references.]
a(n) = A006047(n) - A227428(n).
(End)
From David A. Corneth and Antti Karttunen, Jul 27 2017: (Start)
Based on the first formula above, we have following identities:
a(3n) = a(n).
a(3n+1) = 2*a(n).
a(9n+4) = 4*a(n).
(End)
a(n) = (1/2)*Sum_{k = 0..n} mod(C(n,k) + C(n,k)^2, 3). - Peter Bala, Dec 17 2020

cp's OEIS Frontend

A083093 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 3.

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A194459 Number of entries in the n-th row of Pascal's triangle not divisible by 5.

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A382720 Number of entries in the n-th row of Pascal's triangle not divisible by 7.

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A051638 a(n) = Sum_{k=0..n} (C(n,k) mod 3).

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A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A006048 Number of entries in first n rows of Pascal's triangle not divisible by 3.

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A062296 a(n) = number of entries in n-th row of Pascal's triangle divisible by 3.

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