A131097 Sum of digits of 3-smooth numbers in ternary representation.

1, 2, 1, 2, 2, 4, 1, 2, 4, 2, 4, 1, 4, 2, 4, 2, 4, 4, 1, 4, 2, 6, 4, 2, 4, 4, 1, 4, 4, 2, 6, 4, 2, 8, 4, 4, 1, 4, 4, 2, 8, 6, 4, 2, 8, 4, 4, 10, 1, 4, 4, 2, 8, 6, 4, 10, 2, 8, 4, 4, 10, 1, 4, 4, 8, 2, 8, 6, 4, 10, 2, 8, 4, 10, 4, 10, 1, 4, 4, 8, 2, 8, 6, 16, 4, 10, 2, 8, 4, 10, 4

Offset: 1

Author

Reinhard Zumkeller, Jun 14 2007

Keywords

• nonn
• base
• look

Comments

a(n) = A053735(A003586(n)); values are even iff greater than 1.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Digit Sum
• Eric Weisstein's World of Mathematics, Ternary

Programs

Maple

Res:= NULL: N:= 10^6:
for a from 0 to ilog2(N) do
  for b from 0 do
    v:= 2^a*3^b;
    if v > N then break fi;
    Res:= Res, v;
od od:
TS:= sort([Res]):
map(t -> convert(convert(t,base,3),`+`), TS); # Robert Israel, Oct 08 2018

Python

from sympy import integer_log
from sympy.ntheory import digits
def A131097(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return sum(digits(bisection(f,n,n),3)[1:]) # Chai Wah Wu, Jan 31 2025