A117592 -id:A117592 - cp's OEIS frontend

A081603 Number of 2's in ternary representation of n.

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

Offset: 0

Reinhard Zumkeller, Mar 23 2003

Fixed point of the morphism: 0 ->001; 1 ->112; 2 ->223; 3 ->334, etc., starting from a(0)=0. - Philippe Deléham, Oct 26 2011

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.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A007089, A074940, A005836, A081610, A081611, A117592 (2^a(n)).

a081603 0 = 0
a081603 n = a081603 n' + m `div` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

A081603 := proc(n)
    local a,d ;
    a := 0 ;
    for d in convert(n,base,3) do
        if d= 2 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 12 2016

Table[Count[IntegerDigits[n,3],2],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
Nest[ Flatten[# /. a_Integer -> {a, a, a + 1}] &, {0}, 5] (* Robert G. Wilson v, May 20 2014 *)
DigitCount[Range[0,120],3,2] (* Harvey P. Dale, Jul 10 2019 *)

a(n)=hammingweight(digits(n,3)\2); \\ Ruud H.G. van Tol, Dec 10 2023

from gmpy2 import digits
def A081603(n): return digits(n,3).count('2') # Chai Wah Wu, Dec 05 2024

a(n) = floor(n/2) if n < 3, otherwise a(floor(n/3)) + floor((n mod 3)/2).
A077267(n) + A062756(n) + a(n) = A081604(n);
a(n) = (A053735(n) - A062756(n))/2.

A117942 a(n) = a(3n) = -a(3n+1) = -a(3n+2)/2.

Original entry on oeis.org

1, -1, -2, -1, 1, 2, -2, 2, 4, -1, 1, 2, 1, -1, -2, 2, -2, -4, -2, 2, 4, 2, -2, -4, 4, -4, -8, -1, 1, 2, 1, -1, -2, 2, -2, -4, 1, -1, -2, -1, 1, 2, -2, 2, 4, 2, -2, -4, -2, 2, 4, -4, 4, 8, -2, 2, 4, 2, -2, -4, 4, -4, -8, 2, -2, -4, -2, 2, 4, -4, 4, 8, 4, -4, -8, -4, 4, 8, -8, 8, 16, -1, 1, 2, 1, -1, -2, 2, -2, -4, 1, -1

Offset: 0

Paul Barry, Apr 05 2006

sign

a(n) = a(3n)/a(0) = a(3n+1)/a(1) = a(3n+2)/a(2).

Row sums of A117941.

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

Cf. A117592 (gives the absolute values).

Cf. A062756, A081603.

;; A stand-alone recurrence:
(define (A117942 n) (cond ((zero? n) 1) ((zero? (modulo n 3)) (A117942 (/ n 3))) (else (let ((d (modulo n 3))) (- (* d (A117942 (/ (- n d) 3))))))))
;; An implementation based on a new formula:

(define (A117942 n) (* (A000079 (A081603 n)) (expt -1 (+ (A062756 n) (A081603 n)))))
;; Antti Karttunen, Jul 26 2017

a(n) = 2^A081603(n) * (-1)^(A062756(n)+A081603(n)). - Antti Karttunen, Jul 26 2017

More terms from Antti Karttunen, Jul 26 2017

A338882 Product of the nonzero digits of (n written in base 9).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 2, 4, 6, 8, 10, 12, 14, 16, 3, 3, 6, 9, 12, 15, 18, 21, 24, 4, 4, 8, 12, 16, 20, 24, 28, 32, 5, 5, 10, 15, 20, 25, 30, 35, 40, 6, 6, 12, 18, 24, 30, 36, 42, 48, 7, 7, 14, 21, 28, 35, 42, 49, 56, 8, 8, 16, 24, 32, 40, 48, 56, 64

Offset: 0

Ilya Gutkovskiy, Nov 13 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Product of the nonzero digits of (n written in base k): A000012 (k = 2), A117592 (k = 3), A338854 (k = 4), A338803 (k = 5), A338863 (k = 6), A338880 (k = 7), A338881 (k = 8), this sequence (k = 9), A051801 (k = 10).

Cf. A007095, A309788.

Table[Times @@ DeleteCases[IntegerDigits[n, 9], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5 + 6 x^6 + 7 x^7 + 8 x^8) A[x^9] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Times@@(IntegerDigits[n,9]/.(0->1)),{n,0,80}] (* Harvey P. Dale, Oct 08 2021 *)

a(n) = vecprod(select(x->x, digits(n, 9))); \\ Michel Marcus, Nov 14 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8) * A(x^9).

A338880 Product of the nonzero digits of (n written in base 7).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 2, 2, 4, 6, 8, 10, 12, 3, 3, 6, 9, 12, 15, 18, 4, 4, 8, 12, 16, 20, 24, 5, 5, 10, 15, 20, 25, 30, 6, 6, 12, 18, 24, 30, 36, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 2, 2, 4, 6, 8, 10, 12, 3, 3, 6, 9, 12, 15, 18, 4, 4, 8, 12

Offset: 0

Ilya Gutkovskiy, Nov 13 2020

Alois P. Heinz, Table of n, a(n) for n = 0..9603

Cf. A007093, A051801, A053828, A117592, A309958.

Table[Times @@ DeleteCases[IntegerDigits[n, 7], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5 + 6 x^6) A[x^7] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 7))); \\ Michel Marcus, Nov 14 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6) * A(x^7).

A338803 Product of the nonzero digits of (n written in base 5).

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 2, 2, 4, 6, 8, 3, 3, 6, 9, 12, 4, 4, 8, 12, 16, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 2, 2, 4, 6, 8, 3, 3, 6, 9, 12, 4, 4, 8, 12, 16, 2, 2, 4, 6, 8, 2, 2, 4, 6, 8, 4, 4, 8, 12, 16, 6, 6, 12, 18, 24, 8, 8, 16, 24, 32, 3, 3, 6, 9, 12, 3

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007091, A051801, A117592, A309956.

Table[Times @@ DeleteCases[IntegerDigits[n, 5], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4) A[x^5] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 5))); \\ Michel Marcus, Nov 12 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4) * A(x^5).

A338854 Product of the nonzero digits of (n written in base 4).

Original entry on oeis.org

1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 2, 2, 4, 6, 2, 2, 4, 6, 4, 4, 8, 12, 6, 6, 12, 18, 3, 3, 6, 9, 3, 3, 6, 9, 6, 6, 12, 18, 9, 9, 18, 27, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007090, A051801, A117592, A160382, A160383, A309954.

Table[Times @@ DeleteCases[IntegerDigits[n, 4], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 4))); \\ Michel Marcus, Nov 12 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3) * A(x^4).

a(n) = 2^A160382(n) * 3^A160383(n).

A338863 Product of the nonzero digits of (n written in base 6).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 2, 4, 6, 8, 10, 3, 3, 6, 9, 12, 15, 4, 4, 8, 12, 16, 20, 5, 5, 10, 15, 20, 25, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 2, 4, 6, 8, 10, 3, 3, 6, 9, 12, 15, 4, 4, 8, 12, 16, 20, 5, 5, 10, 15, 20, 25, 2, 2, 4, 6, 8, 10, 2, 2, 4

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007092, A051801, A117592, A309957.

Table[Times @@ DeleteCases[IntegerDigits[n, 6], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5) A[x^6] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 6))); \\ Michel Marcus, Nov 13 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5) * A(x^6).

A338881 Product of the nonzero digits of (n written in base 8).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 13 2020

Cf. A007094, A051801, A117592, A309959.

Table[Times @@ DeleteCases[IntegerDigits[n, 8], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 x^5 + 6 x^6 + 7 x^7) A[x^8] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 8))); \\ Michel Marcus, Nov 14 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7) * A(x^8).

cp's OEIS Frontend

A081603 Number of 2's in ternary representation of n.

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A117942 a(n) = a(3n) = -a(3n+1) = -a(3n+2)/2.

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A338882 Product of the nonzero digits of (n written in base 9).

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A338880 Product of the nonzero digits of (n written in base 7).

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A338803 Product of the nonzero digits of (n written in base 5).

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A338854 Product of the nonzero digits of (n written in base 4).

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A338863 Product of the nonzero digits of (n written in base 6).

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A338881 Product of the nonzero digits of (n written in base 8).

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