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

Author

Gus Wiseman, Dec 23 2020

Keywords

• nonn

Comments

Numbers that can be factored into distinct primes or semiprimes.

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

Examples

			See A339840 for examples.
		

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.

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.

Programs

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/@#]&]!={}&]