A128644 -id:A128644 - cp's OEIS frontend

A037144 Numbers with at most 3 prime factors (counted with multiplicity).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86

Offset: 1

N. J. A. Sloane

nonn

Complement of A033987: A001222(a(n))<=3; A117358(a(n))=1. - Reinhard Zumkeller, Mar 10 2006

Also numbers such that exist permutations of all proper divisors only with coprime adjacent elements: A178254(a(n))>0. - Reinhard Zumkeller, May 24 2010

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

A037143 is a subsequence.

Cf. A033987, A001222, A117358, A128644.

[ n: n in [1..86] | n eq 1 or &+[ t[2]: t in Factorization(n) ] le 3 ]; /* Klaus Brockhaus, Mar 20 2007 */

Select[Range[100],PrimeOmega[#]<4&] (* Harvey P. Dale, Oct 15 2015 *)

is(n)=bigomega(n)<4 \\ Charles R Greathouse IV, Sep 14 2015

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A037144(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-2-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,4)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

More terms from Reinhard Zumkeller, Mar 10 2006

More terms from Klaus Brockhaus, Mar 20 2007

A128645 Number of groups of order A128691(n).

Original entry on oeis.org

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)).

A128694 Number of groups of order A128693(n).

Original entry on oeis.org

2, 1, 5, 2, 1, 2, 2, 1, 15, 2, 4, 1, 1, 2, 2, 2, 4, 1, 2, 5, 1, 2, 1, 55, 5, 1, 2, 13, 2, 2, 1, 2, 2, 1, 2, 1, 4, 2, 5, 1, 2, 1, 2, 5, 1, 14, 2, 2, 4, 1, 16, 1, 2, 2, 1, 2, 5, 2, 2, 261, 2, 1, 15, 1, 2, 1, 2, 4, 49, 1, 2, 1, 2, 4, 5, 2, 2, 5, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 2, 13, 1, 2, 4, 1, 15, 2

Offset: 1

Klaus Brockhaus, Mar 26 2007

nonn

Number of groups for orders of form 3^k*p, where 1 <= k <= 6 and p is a prime different from 3.

The groups of these orders (up to A128693(84005521) = 3221225379 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA.

			A128693(9) = 54 and there are 15 groups of order 54 (A000001(54) = 15), hence a(9) = 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), A128693 (numbers of form 3^k*p, 1<=k<=6, p!=3 prime), A128604 (number of groups for orders that divide p^6, p prime), A128644 (number of groups for orders that have at most 3 prime factors), A128645 (number of groups for orders of form 2^k*p, 1<=k<=8 and p>2 prime).

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..910] | #t eq 2 and ((t[1, 1] eq 2 and t[1, 2] eq 1 and t[2, 1] eq 3 and t[2, 2] le 6) or (t[1, 1] eq 3 and t[1, 2] le 6 and t[2, 2] eq 1)) where t is Factorization(h) ] ];

a(n) = A000001(A128693(n)).

A128704 Number of groups of order A128703(n).

Original entry on oeis.org

2, 1, 1, 5, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 15, 1, 4, 1, 2, 2, 1, 2, 1, 7, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 2, 1, 2, 55, 2, 1, 1, 2, 1, 2, 15, 1, 2, 1, 1, 2, 4, 1, 2, 1, 1, 5, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 1, 21, 1, 1, 1, 2

Offset: 1

Klaus Brockhaus, Mar 26 2007

nonn

Number of groups for orders of form 5^k*p, where 1 <= k <= 5 and p is a prime different from 5.

The groups of these orders (up to A128703(69556991) = 5368708945 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA.

			A128703(20) = 275 and there are 4 groups of order 275 (A000001(275) = 4), hence a(20) = 4.

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), A128703 (numbers of form 5^k*p, 1<=k<=5, p!=5 prime), A128604 (number of groups for orders that divide p^6, p prime), A128644 (number of groups for orders that have at most 3 prime factors), A128645 (number of groups for orders of form 2^k*p, 1<=k<=8, p>2 prime), A128694 (number of groups for orders of form 3^k*p, 1<=k<=6, p!=3 prime).

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..2000] | #t eq 2 and ((t[1, 1] lt 5 and t[1, 2] eq 1 and t[2, 1] eq 5 and t[2, 2] le 5) or (t[1, 1] eq 5 and t[1, 2] le 5 and t[2, 2] eq 1)) where t is Factorization(h) ] ];

a(n) = A000001(A128703(n)).

A128706 Number of groups of order A128705(n).

Original entry on oeis.org

2, 2, 1, 1, 1, 5, 1, 1, 6, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 15, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 19, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 2, 2, 1, 1, 1, 1, 2, 1, 5, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Klaus Brockhaus, Mar 26 2007

nonn

Number of groups for orders of form 7^k*p, where 1 <= k <= 4 and p is a prime different from 7.

The groups of these orders (up to A128705(64633879) = 7516192523 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA.

			A128705(30) = 686 and there are 15 groups of order 686 (A000001(686) = 15), hence a(30) = 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), A128705 (numbers of form 7^k*p, 1<=k<=4, p!=7 prime), A128604 (number of groups for orders that divide p^6, p prime), A128644 (number of groups for orders that have at most 3 prime factors), A128645 (number of groups for orders of form 2^k*p, 1<=k<=8, p>2 prime), A128694 (number of groups for orders of form 3^k*p, 1<=k<=6, p!=3 prime), A128704 (number of groups for orders of form 5^k*p, 1<=k<=5, p!=5 prime).

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..3500] | #t eq 2 and ((t[1, 1] lt 7 and t[1, 2] eq 1 and t[2, 1] eq 7 and t[2, 2] le 4) or (t[1, 1] eq 7 and t[1, 2] le 4 and t[2, 2] eq 1)) where t is Factorization(h) ] ];

a(n) = A000001(A128705(n)).

cp's OEIS Frontend

A037144 Numbers with at most 3 prime factors (counted with multiplicity).

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A128645 Number of groups of order A128691(n).

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A128694 Number of groups of order A128693(n).

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A128704 Number of groups of order A128703(n).

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A128706 Number of groups of order A128705(n).

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