A286586 -id:A286586 - cp's OEIS frontend

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

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

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)).

A286585 a(n) = A053735(A048673(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

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

Cf. A000079, A048673, A053735, A286582, A286583, A286584, A286586.

from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
def a053735(n): return sum(digits(n, 3)[1:])
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 a053735(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A286585 n) (A053735 (A048673 n)))

a(n) = A053735(A048673(n)).

For all n >= 0, a(A000079(n)) = n+1.

A286633 Base-3 {digit+1} product of A254103: a(n) = A006047(A254103(n)).

Original entry on oeis.org

1, 2, 3, 2, 6, 4, 9, 3, 12, 2, 6, 6, 18, 8, 18, 4, 24, 12, 27, 6, 12, 3, 9, 4, 36, 4, 12, 6, 36, 2, 6, 12, 48, 6, 18, 12, 54, 16, 36, 9, 24, 24, 54, 4, 18, 8, 18, 6, 72, 24, 54, 6, 24, 12, 27, 6, 72, 36, 81, 12, 12, 6, 18, 18, 96, 36, 81, 12, 36, 6, 18, 36, 108, 8, 24, 18, 72, 24, 54, 8, 48, 12, 36, 18, 108, 2, 6, 24, 36, 4, 12, 12, 36, 3, 9, 4

Offset: 0

Antti Karttunen, Jun 03 2017

Reflecting the structure of A254103 also this sequence can be represented as a binary tree:
                                     1
                                     |
                  ...................2...................
                 3                                       2
       6......../ \........4                   9......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  12       2           6       6          18       8          18       4
24  12   27 6        12 3     9 4       36  4    12 6       36  2     6 12
etc.

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

Cf. A000079, A006047, A042950, A254103, A286586, A286587, A286632.

from sympy.ntheory.factor_ import digits
from operator import mul
from functools import reduce
def a006047(n):
    d=digits(n, 3)
    return reduce(mul, [1 + d[i] for i in range(1, len(d))])
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n//2) - 1
    else: return (3*(1 + a254103((n - 1)//2)))//2
def a(n): return a006047(a254103(n)) # Indranil Ghosh, Jun 06 2017

(define (A286633 n) (A006047 (A254103 n)))

a(n) = A006047(A254103(n)).

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

A286587 a(n) = A006047(A244154(n)).

Original entry on oeis.org

2, 3, 2, 6, 4, 9, 8, 12, 3, 6, 3, 18, 18, 12, 6, 24, 6, 18, 4, 12, 8, 54, 12, 36, 18, 27, 24, 24, 12, 81, 72, 48, 2, 9, 6, 36, 12, 6, 12, 24, 9, 36, 24, 108, 4, 18, 36, 72, 8, 27, 18, 54, 36, 144, 72, 48, 27, 18, 48, 162, 288, 108, 54, 96, 4, 27, 24, 18, 9, 36, 6, 72, 16, 18, 36, 12, 72, 54, 144, 48, 8, 162, 48, 72, 18, 36, 54, 216, 24

Offset: 0

Antti Karttunen, May 31 2017

nonn

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

Cf. A000225, A005940, A006047, A042950, A244154, A286586.

(define (A286587 n) (A006047 (A244154 n)))

a(n) = A006047(A244154(n)).
a(n) = A286586(A005940(1+n)).
For n >= 0, a(A000225(n)) = A042950(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];

A289623 a(n) = A055396(A048673(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 1, 1, 6, 5, 4, 9, 2, 7, 1, 13, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 16, 8, 1, 2, 10, 2, 30, 2, 3, 14, 3, 1, 23, 1, 17, 1, 1, 2, 4, 18, 1, 1, 4, 1, 1, 1, 35, 1, 15, 11, 1, 1, 1, 1, 3, 1, 1, 1, 1, 21, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 1, 65, 3, 2, 1, 19, 20, 1, 1, 4, 56, 1, 32, 2, 1, 2, 1, 2, 1, 38, 6

Offset: 1

Antti Karttunen, Jul 16 2017

From the scatter plot it can be seen that the terms are grouped into two distinct populations by their magnitude, with significant gap between them.

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

Cf. A048673, A055396, A246263 (the positions of ones).

Cf. also A278224, A286582, A286583, A286584, A286586.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A289623(n) = A055396(A048673(n));

a(n) = A055396(A048673(n)).

cp's OEIS Frontend

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

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

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A286585 a(n) = A053735(A048673(n)).

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A286633 Base-3 {digit+1} product of A254103: a(n) = A006047(A254103(n)).

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A286587 a(n) = A006047(A244154(n)).

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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.

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A289623 a(n) = A055396(A048673(n)).

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