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A100368 Numbers of the form 2^k * p where k > 0 and p is an odd prime.

%I A100368 #41 Dec 21 2020 02:20:27
%S A100368 6,10,12,14,20,22,24,26,28,34,38,40,44,46,48,52,56,58,62,68,74,76,80,
%T A100368 82,86,88,92,94,96,104,106,112,116,118,122,124,134,136,142,146,148,
%U A100368 152,158,160,164,166,172,176,178,184,188,192,194,202,206,208,212,214,218,224
%N A100368 Numbers of the form 2^k * p where k > 0 and p is an odd prime.
%C A100368 Even numbers with 2 distinct prime factors where the odd factor is prime.
%C A100368 A proper subset of A098202. E.g., 210 is not here, but it is there. Also differs from A100367: 36, 100, 108, 196, etc. are missing here. Different also from A036348 because 90 and 180 are not here.
%C A100368 A128691 is a subsequence; A078834(a(n)) = A006530(a(n)). - _Reinhard Zumkeller_, Sep 19 2011
%C A100368 Composite numbers k having the property that the number of divisors of 2k equals the number of divisors of k + 2. All primes satisfy this property. - _Gary Detlefs_, Jan 23 2019
%H A100368 Reinhard Zumkeller, <a href="/A100368/b100368.txt">Table of n, a(n) for n = 1..10000</a>
%F A100368 Numbers of the form 2^k*p where k > 0, p is an odd prime.
%F A100368 a(n) = 2*A038550(n). - _Amiram Eldar_, Dec 21 2020
%p A100368 N:= 1000: # to get all terms <= N
%p A100368 P:= select(isprime, [seq(i,i=3..N/2,2)]):
%p A100368 S:= {seq(seq(2^i*p,i=1..ilog2(N/p)),p=P)}:
%p A100368 sort(convert(S,list)); # _Robert Israel_, Jul 09 2017
%p A100368 with(numtheory): for n from 1 to 224 do if tau(2*n)=tau(n)+2 and not isprime(n) then print(n) fi od # _Gary Detlefs_, Jan 22 2019
%t A100368 <<NumberTheory`NumberTheoryFunctions` p2[x_] :=Part[PrimeFactorList[x], 2]; lf[x_] :=Length[FactorInteger[x]];ta={{0}}; Do[If[Equal[lf[n], 2]&&EvenQ[n]&&IntegerQ[Log[2, n/p2[n]]], ta=Append[ta, n];Print[n]], {n, 1, 256}];ta=Delete[ta, 1]
%t A100368 dpf2Q[n_]:=Module[{f=FactorInteger[n]},Length[f]==2&&f[[1,1]]==2&&f[[-1,2]]==1]; Select[Range[2,250,2],dpf2Q] (* _Harvey P. Dale_, Sep 03 2016 *)
%o A100368 (Haskell)
%o A100368 import Data.Set (singleton, deleteFindMin, insert)
%o A100368 a100368 n = a100368_list !! (n-1)
%o A100368 a100368_list = f (singleton 6) (tail a065091_list) where
%o A100368 f s ps'@(p:ps) | mod m 4 > 0 = m : f (insert (2*p) $ insert (2*m) s') ps
%o A100368 | otherwise = m : f (insert (2*m) s') ps'
%o A100368 where (m,s') = deleteFindMin s
%o A100368 -- _Reinhard Zumkeller_, Sep 19 2011
%o A100368 (PARI) is(n)=n%2==0 && isprime(n>>valuation(n,2)) \\ _Charles R Greathouse IV_, Jul 09 2017
%o A100368 (PARI) list(lim)=my(v=List()); for(k=1,logint(lim\3,2), forprime(p=3,lim>>k, listput(v,p<<k))); Set(v) \\ _Charles R Greathouse IV_, Jul 09 2017
%o A100368 (GAP) a:=Filtered([1..224],n->Tau(2*n)=Tau(n)+2 and not IsPrime(n));; Print(a); # _Muniru A Asiru_, Jan 22 2019
%Y A100368 Cf. A001221, A038550, A098902, A100367, A036348, A100484.
%K A100368 nonn,easy
%O A100368 1,1
%A A100368 _Labos Elemer_, Nov 22 2004
%E A100368 Name edited by _Charles R Greathouse IV_, Jul 09 2017