OFFSET
1,1
COMMENTS
Sequence is a semigroup, since it is closed under multiplication, an associative operation--in fact, it is provably superclosed, i.e., a product of a term in sequence and an arbitrary number is a term in the sequence since the preexisting primes will still be in the new number.
Density: There are 28 terms in the sequence less than 100. Using WolframAlpha, 72% of numbers from 10^20 + 1 through 10^20 + 50 were found to be in the sequence.
Other facts: No primes or prime powers are in the sequence.
Related sequences: Some other sequences that are superclosed semigroups are the counting numbers, the numbers that are not squarefree, and the numbers with initial product in factorization greater than a later prime in the factorization. (See crossrefs.)
EXAMPLE
6 = 2*3 is a term since 2^2 > 3.
1095 = 3*5*73 is a term because 3^2 > 5.
10, 20, and 100 are not terms since 2^2 < 5.
66 = 2*3*11 and 78 = 2*3*13 are terms since 2^2 + 3^2 > 11 and 2^2 + 3^2 = 13.
975560 = 2^3*5*29^3 is a term since 2^2 + 5^2 = 29.
MATHEMATICA
Select[Range@ 203, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^2 > #2 & @@ # &] &] (* Michael De Vlieger, Aug 17 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Richard Locke Peterson, Aug 15 2017
STATUS
approved