%I #43 Jul 13 2024 04:46:47

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

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

%C 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.

%C The next term, if it exists, is > 30000. - _Emeric Deutsch_, Feb 26 2005

%C The next term, if it exists, is > 350000. - _David A. Corneth_, Jul 04 2018

%C The next term, if it exists, is > 2000000. - _Vaclav Kotesovec_, Jul 06 2018

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

%H Robert Heffernan and Des Machale, <a href="https://doi.org/10.33232/BIMS.0093.39.41">The genesis of a conjecture in number theory</a>, Irish Math. Soc. Bulletin Number 93, Summer 2024, 39-41 ISSN 0791-5578. See p. 41.

%e 1 is a term, since p(1) = 1 and 1 divides 1 = 1!.

%e 7 is a term because p(7) = 15 and 15 divides 7! = 5040.

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

%t Do[ If[ Mod[n!, PartitionsP[n]] == 0, Print[n]], {n, 10000}] (* _Robert G. Wilson v_, Nov 23 2004 *)

%t Select[Range[40],Divisible[#!,PartitionsP[#]]&] (* _Harvey P. Dale_, Jan 30 2015 *)

%o (Magma) [ n : n in [1..40] | Factorial(n) mod NumberOfPartitions(n) eq 0 ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

%o (PARI) val(n, p) = my(r=0); while(n,r+=n\=p);r

%o 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

%Y Cf. A000041, A000142, A316530.

%K nonn,nice,more

%O 1,2

%A _Des MacHale_