OFFSET
1,1
COMMENTS
Positive even integers are closed under addition and multiplication. There is no zero and no unit, but the singly even numbers become "primes," and all positive even numbers can be factored into primes.
But unique factorization does not hold. Numbers of the form 4pq, where p is an odd prime and q is any odd integer greater than 1, can be factored as 2(2pq) or as 2p 2q; these are distinct since 2, 2pq, 2p and 2q are all singly even numbers.
For higher k, (2^k)m can have more than two factorizations if Omega(m) >= k, with Omega(n) being the number of prime factors counted with multiplicity (A001222).
REFERENCES
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 18
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
a(n) ~ 4n. In particular, a(n) = 4n + 4n/log n + O(n/log^2 n). - Charles R Greathouse IV, Feb 03 2018
EXAMPLE
36 = 2^2 * 3 * 3. It can be factored into singly even numbers in two different ways: 2 * 18 or 6^2.
60 = 2^2 * 3 * 5. It can be factored into singly even numbers as 2 * 30 or 6 * 10.
MATHEMATICA
Take[DeleteCases[Union[Flatten[Table[2^k * n * Boole[Not[PrimeQ[n]]], {k, 2, 10}, {n, 3, 149, 2}]]], 0], 40]
PROG
(PARI) is(n)=my(k=valuation(n, 2)); k > 1 && !isprime(n>>=k) && n > 1 \\ Charles R Greathouse IV, Dec 28 2012
(PARI) list(lim)=my(v=List()); forcomposite(n=9, lim\4, if(n%2==0, next); my(k=4*n); while(k<=lim, listput(v, k); k<<=1)); Set(v) \\ Charles R Greathouse IV, Feb 03 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alonso del Arte, Dec 28 2012
STATUS
approved