A339889 Products of distinct primes or semiprimes.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70

Offset

1,2

Comments

Numbers that can be factored into distinct primes or semiprimes.

A semiprime (A001358) is a product of any two prime numbers.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Example

See A339840 for examples.

Maple

N:= 100: # for terms <= N
B:= select(t -> numtheory:-bigomega(t) <= 2, {$2..N}):
S:= {1}:
for b in B do
  S:= S union map(`*`, select(`<=`, S, N/b), b)
od:
sort(convert(S, list)); # Robert Israel, Dec 28 2020

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Select[facs[#], UnsameQ@@#&&SubsetQ[{1, 2}, PrimeOmega/@#]&]!={}&]

Crossrefs

See link for additional cross-references.

Allowing only primes gives A005117.

Not allowing squares of primes gives A339741.

Positions of nonzeros in A339839.

Complement of A339840.

A001055 counts factorizations.

A001358 lists semiprimes, with squarefree case A006881.

A320663 counts non-isomorphic multiset partitions into singletons or pairs.

A320732 counts factorizations into primes or semiprimes.

A339742 counts factorizations into distinct primes or squarefree semiprimes.

A339841 have exactly one factorization into primes or semiprimes.

Cf. A002100, A013929, A293511, A320893, A339740, A339846.

Sequence in context: A369939 A178210 A013938 * A023809 A046100 A337138

Adjacent sequences: A339886 A339887 A339888 * A339890 A339891 A339892

Keyword

nonn

Author

Gus Wiseman, Dec 23 2020

Status

approved