OFFSET
1,2
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
MATHEMATICA
Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* Amiram Eldar, Nov 14 2023 *)
PROG
(PARI) for(n=1, 200, t=n/(2^valuation(n, 2)*3^valuation(n, 3)); if((t%4==1), print1(n, ", ")))
(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
(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<<j
if k>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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, May 18 2013
EXTENSIONS
Name clarified by Peter Munn, Nov 10 2023
STATUS
approved