A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 24, 25, 26, 27, 29, 30, 32, 34, 36, 37, 39, 40, 41, 45, 48, 49, 50, 51, 52, 53, 54, 58, 60, 61, 64, 65, 68, 72, 73, 74, 75, 77, 78, 80, 81, 82, 85, 87, 89, 90, 96, 97, 98, 100, 101, 102, 104, 106, 108, 109, 111

Offset: 1

Author

Ralf Stephan, May 18 2013

Keywords

• nonn
• easy

Comments

From Peter Munn, Nov 11 2023: (Start)

Numbers k whose 5-rough part, A065330(k), is congruent to 1 modulo 4.

Contains all nonzero squares.

Positive integers in the multiplicative subgroup of rationals generated by 2, 3, 5 and integers congruent to 1 modulo 12. Thus, the sequence is closed under multiplication and, provided the result is an integer, under division.

This subgroup has index 2 and does not include -1, so is the complement of its negation. In respect of the sequence, the index 2 property implies we can take any absent positive integer m, and divide by m all terms that are multiples of m to get the complementary sequence, A225858.

Likewise, the sequence forms a subgroup of index 2 of the positive integers under the operation A059897(.,.).

(End)

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Nov 14 2023

Links

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

Crossrefs

Complement of A225858.

Cf. A059897, A065330, A225837.

Programs

Magma

[n: n in [1..200] | d mod 4 eq 1 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

Mathematica

Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* Amiram Eldar, Nov 14 2023 *)

PARI

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%4==1),print1(n,",")))

Python

from itertools import count
from sympy import integer_log
def A225857(n):
    def f(x):
        c = n
        for i in range(integer_log(x,3)[0]+1):
            i2 = 3**i
            for j in count(0):
                k = i2<x:
                    break
                m = x//k
                c += (m-7)//12+(m-11)//12+2
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

Extensions

Name clarified by Peter Munn, Nov 10 2023