A188633 Numbers of the form 2^k * m, with k > 1 and m an odd composite number.

36, 60, 72, 84, 100, 108, 120, 132, 140, 144, 156, 168, 180, 196, 200, 204, 216, 220, 228, 240, 252, 260, 264, 276, 280, 288, 300, 308, 312, 324, 336, 340, 348, 360, 364, 372, 380, 392, 396, 400, 408, 420, 432, 440, 444, 456, 460, 468, 476, 480, 484, 492, 500, 504, 516, 520, 528, 532

Offset: 1

Author

Alonso del Arte, Dec 28 2012

Keywords

• nonn
• easy

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).

Examples

			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.
		

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

Crossrefs

Cf. A005843, A016825, A057949.

Programs

Mathematica

Take[DeleteCases[Union[Flatten[Table[2^k * n * Boole[Not[PrimeQ[n]]], {k, 2, 10}, {n, 3, 149, 2}]]], 0], 40]

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

Formula

a(n) ~ 4n. In particular, a(n) = 4n + 4n/log n + O(n/log^2 n). - Charles R Greathouse IV, Feb 03 2018