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

1,1

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

Cf. A005843, A016825, A057949.

Sequence in context: A260131 A320632 A368832 * A328961 A335295 A287862

Adjacent sequences: A188630 A188631 A188632 * A188634 A188635 A188636

Keyword

nonn,easy

Author

Alonso del Arte, Dec 28 2012

Status

approved