A329575 - OEIS

%I #25 Nov 27 2020 04:52:04

%S 3,12,15,21,27,33,39,48,51,57,60,69,75,84,87,93,105,108,111,123,129,

%T 132,135,141,147,156,159,165,177,183,189,192,195,201,204,213,219,228,

%U 231,237,240,243,249,255,267,273,276,285,291,297,300,303,309,321,327,336,339,345

%N Numbers whose smallest Fermi-Dirac factor is 3.

%C 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 Numbers whose squarefree part is divisible by 3 but not 2.

%C 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 Amiram Eldar, <a href="/A329575/b329575.txt">Table of n, a(n) for n = 1..10000</a>

%F A223490(a(n)) = 3.

%F A007913(a(n)) == 3 (mod 6).

%F A059897(2, a(n)) = 2 * a(n).

%F A059897(3, a(n)) * 3 = a(n).

%F {a(n) : n >= 1} = {k : 3 * A307150(k) = 2 * k}.

%F A003159 = {a(n) / 3 : n >= 1} U {a(n) : n >= 1}.

%F A036668 = {a(n) / 3 : n >= 1} U {a(n) * 2 : n >= 1}.

%F A145204 \ {0} = {a(n) : n >= 1} U {a(n) * 2 : n >= 1}.

%e 6 is the product of the following terms of A050376: 2, 3. These terms include 2, so 6 is not in the sequence.

%e 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 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 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 (PARI) isok(m) = core(m) % 6 == 3; \\ _Michel Marcus_, May 01 2020

%Y Cf. A007913, A036668, A050376, A059897, A223490, A307150.

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

%Y Ordered 3rd quadrisection of A052330.

%K nonn

%O 1,1

%A _Peter Munn_, Apr 27 2020