A090184 Number of partitions of the n-th 3-smooth number into parts 2 and 3.

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 22, 25, 28, 33, 37, 41, 43, 49, 55, 65, 73, 82, 86, 97, 109, 122, 129, 145, 163, 171, 193, 217, 244, 257, 289, 325, 342, 365, 385, 433, 487, 513, 577, 649, 683, 730, 769, 865, 973, 1025, 1094, 1153

Offset

1,5

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(2^i * 3^j) = floor(2^(i-1) * 3^(j-1) + 1), i*j>0.

a(n) = A103221(A003586(n)).

Example

n=11: A003586(11) = 2^3 * 3 = 24: 3+3+3+3+3+3+3+3 = 3+3+3+3+3+3+2+2+2 = 3+3+3+3+2+2+2+2+2+2 = 3+3+2+2+2+2+2+2+2+2+2 = 2+2+2+2+2+2+2+2+2+2+2+2: a(11)=5.

Mathematica

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Length[IntegerPartitions[#, All, {2, 3}]]& /@ Select[Range[10000], smooth3Q] (* Jean-François Alcover, Oct 13 2021 *)
With[{nn = 6^5}, Map[Floor[#/2] - Floor[#/3] &, Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}] &[nn] + 2]] (* Michael De Vlieger, Oct 13 2021 *)

Prog

(Python)
from sympy import integer_log
def A090184(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: 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 ((m:=bisection(f, n, n)+2)>>1)-m//3 # Chai Wah Wu, Oct 22 2024

Crossrefs

Cf. A022328, A022329.

Cf. A003586, A008615.

Cf. A103221, A117222, A117220, A117221.

Sequence in context: A067842 A164066 A053251 * A174575 A029057 A087897

Adjacent sequences: A090181 A090182 A090183 * A090185 A090186 A090187

Keyword

nonn

Author

Reinhard Zumkeller, Jan 21 2004

Extensions

Offset changed to 1 by Alois P. Heinz, Oct 15 2021

Status

approved