A128691 -id:A128691 - cp's OEIS frontend

A128645 Number of groups of order A128691(n).

2, 2, 5, 2, 5, 2, 15, 2, 4, 2, 2, 14, 4, 2, 52, 5, 13, 2, 2, 5, 2, 4, 52, 2, 2, 12, 4, 2, 231, 14, 2, 43, 5, 2, 2, 4, 2, 15, 2, 2, 5, 12, 2, 238, 5, 2, 4, 42, 2, 12, 4, 1543, 2, 2, 2, 51, 5, 2, 2, 197, 2, 14, 4, 5, 12, 2, 2, 4, 54, 2, 2, 4, 5, 14, 2, 2, 42, 2, 4, 1640, 2, 15, 4, 2, 12, 2, 195, 5, 2

Offset: 1

Klaus Brockhaus, Mar 21 2007

nonn

Number of groups whose order is of form 2^k*p, where 1 <= k <= 8 and p is a prime > 2.

The groups of these orders (up to A128691(112490698) = 2147483636 in version V2.13-4) form a class contained in the Small Groups Library of Magma.

			A128691(7) = 24 and there are 15 groups of order 24 (A000001(24) = 15), hence a(7) = 15.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
Magma Computational Algebra System, Documentation, see Database of Small Groups.

Cf. A000001 (number of groups of order n), A128691 (numbers of form 2^k*p, 1<=k<=8, p > 2 prime), A128604 (number of groups whose order divides p^6 for p a prime), A128644 (number of groups whose order has at most 3 prime factors).

D := SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [ h: h in [1..360] | #t eq 2 and t[1, 1] eq 2 and t[1, 2] le 8 and t[2, 2] eq 1 where t is Factorization(h) ] ];

a(n) = A000001(A128691(n)).

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

Original entry on oeis.org

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

Labos Elemer, Nov 22 2004

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

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

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

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

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

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 *)
```

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

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

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

Name edited by Charles R Greathouse IV, Jul 09 2017

cp's OEIS Frontend

A128645 Number of groups of order A128691(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

PARI

Formula

Extensions