A046668 Numbers n such that partition function p(n) divides n!.

1, 2, 3, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 24, 28, 32, 33, 39

Offset

1,2

Comments

The symmetric group has p(n) conjugacy classes and order n! The sequence arose in a search for groups G which satisfy Pr(G) = k(G)/|G| = 1/t, for integer t, where G has k(G) conjugacy classes.

The next term, if it exists, is > 30000. - Emeric Deutsch, Feb 26 2005

The next term, if it exists, is > 350000. - David A. Corneth, Jul 04 2018

The next term, if it exists, is > 2000000. - Vaclav Kotesovec, Jul 06 2018

References

Commutativity and Generalizations in Finite Groups; Aine NiShe, Ph.D. thesis in preparation.

Links

Table of n, a(n) for n=1..21.

Example

1 is a term, since p(1) = 1 and 1 divides 1 = 1!.
7 is a term because p(7) = 15 and 15 divides 7! = 5040.

Maple

with(combinat): p:=proc(n) if type(n!/numbpart(n), integer) then n fi end; seq(p(n), n=1..300); # Emeric Deutsch

Mathematica

Do[ If[ Mod[n!, PartitionsP[n]] == 0, Print[n]], {n, 10000}] (* Robert G. Wilson v, Nov 23 2004 *)
Select[Range[40], Divisible[#!, PartitionsP[#]]&] (* Harvey P. Dale, Jan 30 2015 *)

Prog

(Magma) [ n : n in [1..40] | Factorial(n) mod NumberOfPartitions(n) eq 0 ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(PARI) val(n, p) = my(r=0); while(n, r+=n\=p); r
is(n) = qp = numbpart(n); forprime(p = 2, n, if(val(n, p) < valuation(qp, p), return(0)); qp/=p^valuation(qp, p)); qp==1 \\ David A. Corneth, Jul 04 2018

Crossrefs

Cf. A000041, A000142, A316530.

Sequence in context: A323384 A349641 A140221 * A047533 A060525 A152863

Adjacent sequences: A046665 A046666 A046667 * A046669 A046670 A046671

Keyword

nonn,nice,more

Author

Des MacHale

Status

approved