OFFSET
1,1
COMMENTS
Every positive integer is the product of a unique subset of the terms of A050376 (sometimes called Fermi-Dirac primes). This sequence lists the numbers where the relevant subset includes 3 but not 2.
Numbers whose squarefree part is divisible by 3 but not 2.
Positive multiples of 3 that survive sieving by the rule: if m appears then 2m, 3m and 6m do not. Asymptotic density is 1/6.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.
FORMULA
A223490(a(n)) = 3.
A007913(a(n)) == 3 (mod 6).
A059897(2, a(n)) = 2 * a(n).
A059897(3, a(n)) * 3 = a(n).
{a(n) : n >= 1} = {k : 3 * A307150(k) = 2 * k}.
A003159 = {a(n) / 3 : n >= 1} U {a(n) : n >= 1}.
A036668 = {a(n) / 3 : n >= 1} U {a(n) * 2 : n >= 1}.
A145204 \ {0} = {a(n) : n >= 1} U {a(n) * 2 : n >= 1}.
a(n) = 3*A339690(n). - Chai Wah Wu, Apr 10 2025
EXAMPLE
6 is the product of the following terms of A050376: 2, 3. These terms include 2, so 6 is not in the sequence.
12 is the product of the following terms of A050376: 3, 4. These terms include 3, but not 2, so 12 is in the sequence.
20 is the product of the following terms of A050376: 4, 5. These terms do not include 3, so 20 is not in the sequence.
MATHEMATICA
f[p_, e_] := p^(2^IntegerExponent[e, 2]); fdmin[n_] := Min @@ f @@@ FactorInteger[n]; Select[Range[350], fdmin[#] == 3 &] (* Amiram Eldar, Nov 27 2020 *)
PROG
(PARI) isok(m) = core(m) % 6 == 3; \\ Michel Marcus, May 01 2020
(Python)
from itertools import count
from sympy import integer_log
def A329575(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 3*bisection(f, n, n) # Chai Wah Wu, Apr 10 2025
CROSSREFS
Ordered 3rd quadrisection of A052330.
KEYWORD
nonn
AUTHOR
Peter Munn, Apr 27 2020
STATUS
approved