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
Robert Heffernan and Des Machale, The genesis of a conjecture in number theory, Irish Math. Soc. Bulletin Number 93, Summer 2024, 39-41 ISSN 0791-5578. See p. 41.
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
KEYWORD
nonn,nice,more
AUTHOR
STATUS
approved