A135850 Numbers m such that there are precisely 6 groups of order m.

42, 78, 110, 114, 147, 186, 222, 225, 258, 310, 366, 402, 406, 410, 438, 474, 506, 507, 525, 582, 602, 610, 618, 654, 710, 735, 762, 834, 906, 942, 975, 978, 994, 1010, 1083, 1086, 1089, 1158, 1194, 1266, 1310, 1338, 1374, 1378, 1425, 1446, 1474, 1510, 1582

Offset

1,1

Comments

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

Links

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..1099 (terms 1..91 from Muniru A Asiru).

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.

Index entries for sequences related to groups

Formula

Sequence is { m | A000001(m) = 6 }. - Muniru A Asiru, Nov 04 2017

Example

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

Mathematica

Select[Range[10^4], FiniteGroupCount[#] == 6 &] (* Robert Price, May 23 2019 *)

Prog

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

Crossrefs

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).

Sequence in context: A072326 A068700 A303283 * A250381 A153644 A172437

Adjacent sequences: A135847 A135848 A135849 * A135851 A135852 A135853

Keyword

nonn

Author

N. J. A. Sloane, based on a suggestion from Neven Juric, Mar 08 2008

Status

approved