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

1,1

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

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}.

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

Crossrefs

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

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

Ordered 3rd quadrisection of A052330.

Sequence in context: A044828 A248105 A269315 * A022380 A331074 A290593

Adjacent sequences: A329572 A329573 A329574 * A329576 A329577 A329578

Keyword

nonn

Author

Peter Munn, Apr 27 2020

Status

approved