A329575 Numbers whose smallest Fermi-Dirac factor is 3.

3, 12, 15, 21, 27, 33, 39, 48, 51, 57, 60, 69, 75, 84, 87, 93, 105, 108, 111, 123, 129, 132, 135, 141, 147, 156, 159, 165, 177, 183, 189, 192, 195, 201, 204, 213, 219, 228, 231, 237, 240, 243, 249, 255, 267, 273, 276, 285, 291, 297, 300, 303, 309, 321, 327, 336, 339, 345

Offset: 1

Author

Peter Munn, Apr 27 2020

Keywords

• nonn

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Crossrefs

Cf. A007913, A036668, A050376, A059897, A223490, A307150, A339690.

Intersection of any 2 of A003159, A145204 and A325424; also subsequence of A028983.

Ordered 3rd quadrisection of A052330.

Programs

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 *)

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<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

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