A051178 Numbers k such that k divides the number of divisors of k!.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 45, 48, 52, 54, 56, 60, 64, 66, 70, 72, 76, 78, 80, 82, 84, 90, 96, 100, 102, 105, 108, 110, 112, 114, 120, 125, 126, 128, 130, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168

Offset: 1

Author

Leroy Quet

Keywords

• nonn
• nice

Comments

This sequence is not closed under multiplication. For example, 2*52=104 and 2*82=164 are not terms. - Ivan N. Ianakiev, Jan 23 2023

Examples

			6 is a term because the number of divisors of 6! is 30, which is divisible by 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A000005, A000142, A027423, A277166.

Programs

Haskell

a051178 n = a051178_list !! (n-1)
a051178_list = filter (\x -> a027423 x `mod` x == 0) [1..]
-- Reinhard Zumkeller, Feb 27 2013

Mathematica

ok[n_] := Divisible[ DivisorSigma[0, n!], n]; Select[ Range[200], ok] (* Jean-François Alcover, Dec 08 2011 *)
Select[Range[200],Mod[DivisorSigma[0,#!],#]==0&] (* Harvey P. Dale, Mar 13 2023 *)

PARI

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(s=1); forprime(p=2,n, s*=valp(n,p)+1; s%=n; if(s==0, return(1))); n==1 \\ Charles R Greathouse IV, Nov 04 2016

Formula

It seems that a(n) is asymptotic to c*n with c=3.2..... - Benoit Cloitre, Sep 03 2002

A027423(a(n)) mod a(n) = 0. - Reinhard Zumkeller, Feb 27 2013

No member > 2 is prime. - Charlie Neder, Dec 23 2018