A023694 -id:A023694 - cp's OEIS frontend

A062756 Number of 1's in ternary (base-3) expansion of n.

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

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001

Fixed point of the morphism: 0 ->010; 1 ->121; 2 ->232; ...; n -> n(n+1)n, starting from a(0)=0. - Philippe Deléham, Oct 25 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.
Michael Gilleland, Some Self-Similar Integer Sequences
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Kevin Ryde, Iterations of the Terdragon Curve, see index "dir".
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A080846, A343785 (first differences).
Cf. A081606 (indices of !=0).
Indices of terms 0..6: A005823, A023692, A023693, A023694, A023695, A023696, A023697.
Numbers of: A077267 (0's), A081603 (2's), A160384 (1's+2's).
Other bases: A000120, A160381, A268643.

a062756 0 = 0
a062756 n = a062756 n' + m `mod` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

Table[Count[IntegerDigits[i, 3], 1], {i, 0, 200}]
Nest[Join[#, # + 1, #] &, {0}, 5] (* IWABUCHI Yu(u)ki, Sep 08 2012 *)

a(n)=if(n<1,0,a(n\3)+(n%3)%2) \\ Paul D. Hanna, Feb 24 2006

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

from sympy.ntheory import digits
def A062756(n): return digits(n,3)[1:].count(1) # Chai Wah Wu, Dec 23 2022

a(0) = 0, a(3n) = a(n), a(3n+1) = a(n)+1, a(3n+2) = a(n). - Vladeta Jovovic, Jul 18 2001
G.f.: (Sum_{k>=0} x^(3^k)/(1+x^(3^k)+x^(2*3^k)))/(1-x). In general, the generating function for the number of digits equal to d in the base b representation of n (0 < d < b) is (Sum_{k>=0} x^(d*b^k)/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005 [For d=0, use the above formula with d=b: (Sum_{k>=0} x^(b^(k+1))/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x), adding 1 if you consider the representation of 0 to have one zero digit.]
a(n) = a(floor(n/3)) + (n mod 3) mod 2. - Paul D. Hanna, Feb 24 2006

More terms from Vladeta Jovovic, Jul 18 2001

A226636 Numbers whose base-3 sum of digits is 3.

Original entry on oeis.org

5, 7, 11, 13, 15, 19, 21, 29, 31, 33, 37, 39, 45, 55, 57, 63, 83, 85, 87, 91, 93, 99, 109, 111, 117, 135, 163, 165, 171, 189, 245, 247, 249, 253, 255, 261, 271, 273, 279, 297, 325, 327, 333, 351, 405, 487, 489, 495, 513, 567, 731, 733, 735, 739, 741, 747, 757

Offset: 1

Tom Edgar, Aug 31 2013

All of the entries are odd.
Subsequence of A005408. - Michel Marcus, Sep 03 2013
In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

			The ternary expansion of 5 is (1,2), which has sum of digits 3.
The ternary expansion of 31 is (1,0,0,2), which has sum of digits 3.
10 is not on the list since the ternary expansion of 10 is (1,0,1), which has sum of digits 2 not 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A018900, A023694, A055235, A187813.

Cf. A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9), A052224 (b = 10).

N:= 10: # for all terms < 3^(N+1)
[seq(seq(seq(3^a+3^b+3^c, c=0..`if`(b=a, b-1,b)),b = 0..a),a=0..N)]; # Robert Israel, Jun 05 2018

Select[Range@ 757, Total@ IntegerDigits[#, 3] == 3 &] (* Michael De Vlieger, Dec 23 2016 *)

select( is(n)=sumdigits(n,3)==3, [1..999]) \\ M. F. Hasler, Dec 23 2016

from itertools import islice
def nextsod(n, base):
    c, b, w = 0, base, 0
    while True:
        d = n%b
        if d+1 < b and c:
            return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
        c += d; n //= b; w += 1
def A226636gen(sod=3, base=3): # generator of terms for any sod, base
    an = (sod%(base-1)+1)*base**(sod//(base-1))-1
    while True: yield an; an = nextsod(an, base)
print(list(islice(A226636gen(), 57))) # Michael S. Branicky, Jul 10 2022, generalizing the code by M. F. Hasler in A052224

[i for i in [0..1000] if sum(Integer(i).digits(base=3))==3]

a(k^3/6 + k^2 + 5*k/6 + j) = 3^(k+1) + A055235(j-1) for 1 <= j <= k^2/2+5*k/2+2. - Robert Israel, Jun 05 2018

A281004 Numbers with exactly 3 ones in both binary and ternary representations.

Original entry on oeis.org

13, 37, 41, 49, 67, 97, 131, 133, 145, 193, 259, 265, 273, 289, 385, 517, 529, 577, 1027, 1029, 1033, 1041, 1153, 1281, 2053, 2057, 4101, 4105, 4113, 4129, 4161, 6145, 8195, 8197, 8209, 8225, 8257, 8321, 8449, 8705, 10241, 16449, 17409, 18433, 20481, 24577, 32771, 32777, 32785, 32801, 32833, 32897

Offset: 1

Robert Israel, Jan 12 2017

Intersection of A014311 and A023694.
All terms are odd, since n == A062756(n) (mod 2).
It is likely that a(136) = 1099528404993 is the last term.  The next term, if any, is greater than 10^200.

			a(4) = 49 = 110001_2 = 1211_3.

Robert Israel, Table of n, a(n) for n = 1..136

Cf. A014311, A023694, A062756. Contains A280997.

R:= NULL: count:= 0:
for a from 2 while count < 136 do
  for b from 1 to a-1  do
    p:= 2^a + 2^b + 1;
    if numboccur(1, convert(p,base,3)) = 3 then
      count:= count+1;
      R:= R, p;
    fi
od od:
R;

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A062756 Number of 1's in ternary (base-3) expansion of n.

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A226636 Numbers whose base-3 sum of digits is 3.

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A281004 Numbers with exactly 3 ones in both binary and ternary representations.

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Maple