cp's OEIS Frontend

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

%I A188633 #24 Feb 03 2018 02:34:17
%S A188633 36,60,72,84,100,108,120,132,140,144,156,168,180,196,200,204,216,220,
%T A188633 228,240,252,260,264,276,280,288,300,308,312,324,336,340,348,360,364,
%U A188633 372,380,392,396,400,408,420,432,440,444,456,460,468,476,480,484,492,500,504,516,520,528,532
%N A188633 Numbers of the form 2^k * m, with k > 1 and m an odd composite number.
%C A188633 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.
%C A188633 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.
%C A188633 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).
%D A188633 Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 18
%H A188633 Charles R Greathouse IV, <a href="/A188633/b188633.txt">Table of n, a(n) for n = 1..10000</a>
%F A188633 a(n) ~ 4n. In particular, a(n) = 4n + 4n/log n + O(n/log^2 n). - _Charles R Greathouse IV_, Feb 03 2018
%e A188633 36 = 2^2 * 3 * 3. It can be factored into singly even numbers in two different ways: 2 * 18 or 6^2.
%e A188633 60 = 2^2 * 3 * 5. It can be factored into singly even numbers as 2 * 30 or 6 * 10.
%t A188633 Take[DeleteCases[Union[Flatten[Table[2^k * n * Boole[Not[PrimeQ[n]]], {k, 2, 10}, {n, 3, 149, 2}]]], 0], 40]
%o A188633 (PARI) is(n)=my(k=valuation(n,2));k > 1 && !isprime(n>>=k) && n > 1 \\ _Charles R Greathouse IV_, Dec 28 2012
%o A188633 (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
%Y A188633 Cf. A005843, A016825, A057949.
%K A188633 nonn,easy
%O A188633 1,1
%A A188633 _Alonso del Arte_, Dec 28 2012