A135850 - OEIS

%I #35 May 13 2023 23:50:21

%S 42,78,110,114,147,186,222,225,258,310,366,402,406,410,438,474,506,

%T 507,525,582,602,610,618,654,710,735,762,834,906,942,975,978,994,1010,

%U 1083,1086,1089,1158,1194,1266,1310,1338,1374,1378,1425,1446,1474,1510,1582

%N Numbers m such that there are precisely 6 groups of order m.

%C Let gnu(n) = A000001(n) denote the "group number of n" defined in A000001 or in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), then the sequence n -> gnu(a(n)) -> gnu(gnu(a(n))) -> gnu(gnu(gnu(a(n)))) consists of 1's. - _Muniru A Asiru_, Nov 19 2017

%H Jorge R. F. F. Lopes, <a href="/A135850/b135850.txt">Table of n, a(n) for n = 1..1099</a> (terms 1..91 from Muniru A Asiru).

%H J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf"> Counting groups: gnus, moas and other exotica</a>, Math. Intell., Vol. 30, No. 2, Spring 2008.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F Sequence is { m | A000001(m) = 6 }. - _Muniru A Asiru_, Nov 04 2017

%e For m = 42, the 6 groups of order 42 are (C7 : C3) : C2, C2 x (C7 : C3), C7 x S3, C3 x D14, D42, C42 and for n = 78 the 6 groups of order 78 are (C13 : C3) : C2, C2 x (C13 : C3), C13 x S3, C3 x D26, D78, C78 where C, D mean Cyclic, Dihedral groups of the stated order and S is the Symmetric group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 04 2017

%t Select[Range[10^4], FiniteGroupCount[#] == 6 &] (* _Robert Price_, May 23 2019 *)

%o (GAP) A135850 := Filtered([1..2015], n -> NumberSmallGroups(n) = 6); # _Muniru A Asiru_, Nov 04 2017

%Y Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), this sequence (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

%K nonn

%O 1,1

%A _N. J. A. Sloane_, based on a suggestion from Neven Juric, Mar 08 2008