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A100368 Numbers of the form 2^k * p where k > 0 and p is an odd prime. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

FORMULA

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

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

<<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]

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 *)

PROG

(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

(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<<k))); Set(v) \\ Charles R Greathouse IV, Jul 09 2017

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

CROSSREFS

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

Sequence in context: A295076 A088829 A036348 * A128691 A028919 A325231

Adjacent sequences:  A100365 A100366 A100367 * A100369 A100370 A100371

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Nov 22 2004

EXTENSIONS

Name edited by Charles R Greathouse IV, Jul 09 2017

STATUS

approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)