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

6, 10, 12, 14, 20, 22, 24, 26, 28, 34, 38, 40, 44, 46, 48, 52, 56, 58, 62, 68, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 122, 124, 134, 136, 142, 146, 148, 152, 158, 160, 164, 166, 172, 176, 178, 184, 188, 192, 194, 202, 206, 208, 212, 214, 218, 224

Offset: 1

Author

Labos Elemer, Nov 22 2004

Keywords

• nonn
• easy

Comments

Even numbers with 2 distinct prime factors where the odd factor is prime.

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.

A128691 is a subsequence; A078834(a(n)) = A006530(a(n)). - Reinhard Zumkeller, Sep 19 2011

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A038550, A098902, A100367, A036348, A100484.

Programs

GAP

a:=Filtered([1..224],n->Tau(2*n)=Tau(n)+2 and not IsPrime(n));; Print(a); # Muniru A Asiru, Jan 22 2019

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a100368 n = a100368_list !! (n-1)
a100368_list = f (singleton 6) (tail a065091_list) where
f s ps'@(p:ps) | mod m 4 > 0 = m : f (insert (2*p) $ insert (2*m) s') ps
| otherwise = m : f (insert (2*m) s') ps'
where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 19 2011

Maple

N:= 1000: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..N/2,2)]):
S:= {seq(seq(2^i*p,i=1..ilog2(N/p)),p=P)}:
sort(convert(S,list)); # Robert Israel, Jul 09 2017
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

Mathematica

<Harvey P. Dale, Sep 03 2016 *)

PARI

is(n)=n%2==0 && isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Jul 09 2017

PARI

list(lim)=my(v=List()); for(k=1,logint(lim\3,2), forprime(p=3,lim>>k, listput(v,p<Charles R Greathouse IV, Jul 09 2017

Formula

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

a(n) = 2*A038550(n). - Amiram Eldar, Dec 21 2020

Extensions

Name edited by Charles R Greathouse IV, Jul 09 2017