cp's OEIS Frontend

A329575 Numbers whose smallest Fermi-Dirac factor is 3.

%I A329575 #37 Apr 22 2025 05:08:46
%S A329575 3,12,15,21,27,33,39,48,51,57,60,69,75,84,87,93,105,108,111,123,129,
%T A329575 132,135,141,147,156,159,165,177,183,189,192,195,201,204,213,219,228,
%U A329575 231,237,240,243,249,255,267,273,276,285,291,297,300,303,309,321,327,336,339,345
%N A329575 Numbers whose smallest Fermi-Dirac factor is 3.
%C A329575 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.
%C A329575 Numbers whose squarefree part is divisible by 3 but not 2.
%C A329575 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.
%H A329575 Amiram Eldar, <a href="/A329575/b329575.txt">Table of n, a(n) for n = 1..10000</a>
%H A329575 Jan Snellman, <a href="https://arxiv.org/abs/2504.02795">Greedy Regular Convolutions</a>, arXiv:2504.02795 [math.NT], 2025.
%F A329575 A223490(a(n)) = 3.
%F A329575 A007913(a(n)) == 3 (mod 6).
%F A329575 A059897(2, a(n)) = 2 * a(n).
%F A329575 A059897(3, a(n)) * 3 = a(n).
%F A329575 {a(n) : n >= 1} = {k : 3 * A307150(k) = 2 * k}.
%F A329575 A003159 = {a(n) / 3 : n >= 1} U {a(n) : n >= 1}.
%F A329575 A036668 = {a(n) / 3 : n >= 1} U {a(n) * 2 : n >= 1}.
%F A329575 A145204 \ {0} = {a(n) : n >= 1} U {a(n) * 2 : n >= 1}.
%F A329575 a(n) = 3*A339690(n). - _Chai Wah Wu_, Apr 10 2025
%e A329575 6 is the product of the following terms of A050376: 2, 3. These terms include 2, so 6 is not in the sequence.
%e A329575 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.
%e A329575 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.
%t A329575 f[p_, e_] := p^(2^IntegerExponent[e, 2]); fdmin[n_] := Min @@ f @@@ FactorInteger[n]; Select[Range[350], fdmin[#] == 3 &] (* _Amiram Eldar_, Nov 27 2020 *)
%o A329575 (PARI) isok(m) = core(m) % 6 == 3; \\ _Michel Marcus_, May 01 2020
%o A329575 (Python)
%o A329575 from itertools import count
%o A329575 from sympy import integer_log
%o A329575 def A329575(n):
%o A329575     def bisection(f,kmin=0,kmax=1):
%o A329575         while f(kmax) > kmax: kmax <<= 1
%o A329575         kmin = kmax >> 1
%o A329575         while kmax-kmin > 1:
%o A329575             kmid = kmax+kmin>>1
%o A329575             if f(kmid) <= kmid:
%o A329575                 kmax = kmid
%o A329575             else:
%o A329575                 kmin = kmid
%o A329575         return kmax
%o A329575     def f(x):
%o A329575         c = n+x
%o A329575         for i in range(integer_log(x,9)[0]+1):
%o A329575             i2 = 9**i
%o A329575             for j in count(0,2):
%o A329575                 k = i2<<j
%o A329575                 if k>x:
%o A329575                     break
%o A329575                 m = x//k
%o A329575                 c -= (m-1)//6+(m-5)//6+2
%o A329575         return c
%o A329575     return 3*bisection(f,n,n) # _Chai Wah Wu_, Apr 10 2025
%Y A329575 Cf. A007913, A036668, A050376, A059897, A223490, A307150, A339690.
%Y A329575 Intersection of any 2 of A003159, A145204 and A325424; also subsequence of A028983.
%Y A329575 Ordered 3rd quadrisection of A052330.
%K A329575 nonn
%O A329575 1,1
%A A329575 _Peter Munn_, Apr 27 2020