OFFSET
1,2
COMMENTS
Positive integers that survive sieving by the rule: if m appears then 2m, 3m and 6m do not.
Numbers whose squarefree part is congruent to 1 or 5 modulo 6.
Closed under multiplication.
Term by term, the sequence is one half of its complement within A007417, one third of its complement within A003159, and one sixth of its complement within A036668.
Asymptotic density is 1/2.
The set of all a(n) has maximal lower density (1/2) among sets S such that S, 2S, and 3S are disjoint.
Numbers which do not have 2 or 3 in their Fermi-Dirac factorization. Thus each term is a product of a unique subset of A050376 \ {2,3}.
It follows that the sequence is closed with respect to the commutative binary operation A059897(.,.), forming a subgroup of the positive integers considered as a group under A059897. It is the subgroup generated by A050376 \ {2,3}. A003159, A007417 and A036668 correspond to the nontrivial subgroups of its quotient group. It is the lexicographically earliest ordered transversal of the subgroup {1,2,3,6}, which in ordered form is the lexicographically earliest subgroup of order 4.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.
Eric Weisstein's World of Mathematics, Group.
Eric Weisstein's World of Mathematics, Squarefree Part.
FORMULA
{a(n) : n >= 1} = {m : A307150(m) = 6m, m >= 0}.
{a(n) : n >= 1} = {k : k = A052330(4m), m >= 0}.
A329575(n) = a(n) * 3.
{A036668(n) : n >= 0} = {a(n) : n >= 1} U {6 * a(n) : n >= 1}.
{A003159(n) : n >= 1} = {a(n) : n >= 1} U {3 * a(n) : n >= 1}.
{A007417(n) : n >= 1} = {a(n) : n >= 1} U {2 * a(n) : n >= 1}.
a(n) ~ 2n.
EXAMPLE
Numbers are removed by the sieve only due to the presence of a smaller number, so 1 is in the sequence as the smallest positive integer. The sieve removes 2, as it is twice 1, which is in the sequence; so 2 is not in the sequence. The sieve removes 3, as it is three times 1, which is in the sequence, so 3 is not in the sequence. There are no integers m for which 3m = 4 or 6m = 4; 2m = 4 for m = 2, but 2 is not in the sequence; so the sieve does not remove 4, so 4 is in the sequence.
MAPLE
filter:= n -> padic:-ordp(n, 2)::even and padic:-ordp(n, 3)::even;
select(filter, [$1..200]); # Robert Israel, Dec 24 2025
MATHEMATICA
Select[Range[117], EvenQ[IntegerExponent[#, 2]] && EvenQ[IntegerExponent[#, 3]] &]
f[p_, e_] := p^Mod[e, 2]; core[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[121], CoprimeQ[core[#], 6] &] (* Amiram Eldar, Feb 06 2021 *)
PROG
(PARI) isok(m) = core(m) % 6 == 1 || core(m) % 6 == 5;
(Python)
from itertools import count
from sympy import integer_log
def A339690(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):
c = n+x
for i in range(integer_log(x, 9)[0]+1):
i2 = 9**i
for j in count(0, 2):
k = i2<<j
if k>x:
break
m = x//k
c -= (m-1)//6+(m-5)//6+2
return c
return bisection(f, n, n) # Chai Wah Wu, Feb 14 2025
KEYWORD
nonn
AUTHOR
Griffin N. Macris, Dec 13 2020, and Peter Munn, Feb 03 2021
STATUS
approved