A225837 Numbers of form 2^i*3^j*(6k+1), i, j, k >= 0.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 25, 26, 27, 28, 31, 32, 36, 37, 38, 39, 42, 43, 48, 49, 50, 52, 54, 55, 56, 57, 61, 62, 63, 64, 67, 72, 73, 74, 75, 76, 78, 79, 81, 84, 85, 86, 91, 93, 96, 97, 98, 100, 103, 104, 108, 109, 110, 111, 112

Offset: 1

Author

Ralf Stephan, May 16 2013

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022

From Peter Munn, Nov 16 2023: (Start)

Contains all nonzero squares.

Dividing by 5 the terms that are multiples of 5 gives its complement, A225838.

(A352272, 2*A352272, 3*A352272, 6*A352272) is a partition of the terms.

The terms form a subgroup of the positive integers under the operation A059897(.,.) and are the positive integers in an index 2 multiplicative subgroup of rationals that is generated by 2, 3 and integers congruent to 1 modulo 6. See A225857 and A352272 for further information about such subgroups.

(End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A225838.

Subsequences: A003136\{0}, A083854\{0}, A260488\{0}, A352272.

Symmetric difference of A026225 and A036554; of A036668 and A189716.

Cf. A016921, A059897, A225857.

Programs

Magma

[n: n in [1..200] | IsOne(d mod 6) where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

Mathematica

mx = 122; t = {}; Do[n = 2^i*3^j (6 k + 1); If[n <= mx, AppendTo[t, n]], {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, mx/6}]; Union[t] (* T. D. Noe, May 16 2013 *)

PARI

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==1),print1(n,",")))

Python

from sympy import integer_log
def A225837(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>>j)+5)//6 for i in range(integer_log(x,3)[0]+1) for j in range((x//3**i).bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025