A128704 -id:A128704 - cp's OEIS frontend

A128703 Numbers of the form 5^k*p, where 1 <= k <= 5 and p is a prime different from 5.

10, 15, 35, 50, 55, 65, 75, 85, 95, 115, 145, 155, 175, 185, 205, 215, 235, 250, 265, 275, 295, 305, 325, 335, 355, 365, 375, 395, 415, 425, 445, 475, 485, 505, 515, 535, 545, 565, 575, 635, 655, 685, 695, 725, 745, 755, 775, 785, 815, 835, 865, 875, 895, 905

Offset: 1

Klaus Brockhaus, Mar 26 2007

nonn

Auxiliary sequence for A128704 which gives the number of groups of order a(n).

			375 = 5^3*3 is a term.

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

Cf. A128704, A000351 (powers of 5).

[ n: n in [1..910] | #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(n) ];

With[{upto=1000},Select[Union[Flatten[5^Range[5] #&/@Drop[Prime[ Range[ PrimePi[ Ceiling[upto/5]]]],{3}]]],#<=upto&]] (* Harvey P. Dale, Jul 27 2011 *)

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

A128703 Numbers of the form 5^k*p, where 1 <= k <= 5 and p is a prime different from 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica