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Numbers k such that k divides the number of divisors of k!.
3

%I #48 Mar 13 2023 11:55:27

%S 1,2,4,6,8,10,12,16,18,20,24,27,28,30,32,36,40,42,45,48,52,54,56,60,

%T 64,66,70,72,76,78,80,82,84,90,96,100,102,105,108,110,112,114,120,125,

%U 126,128,130,132,135,136,140,144,150,152,156,160,162,168

%N Numbers k such that k divides the number of divisors of k!.

%C 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

%H Amiram Eldar, <a href="/A051178/b051178.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%F It seems that a(n) is asymptotic to c*n with c=3.2..... - _Benoit Cloitre_, Sep 03 2002

%F A027423(a(n)) mod a(n) = 0. - _Reinhard Zumkeller_, Feb 27 2013

%F No member > 2 is prime. - _Charlie Neder_, Dec 23 2018

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

%t ok[n_] := Divisible[ DivisorSigma[0, n!], n]; Select[ Range[200], ok] (* _Jean-François Alcover_, Dec 08 2011 *)

%t Select[Range[200],Mod[DivisorSigma[0,#!],#]==0&] (* _Harvey P. Dale_, Mar 13 2023 *)

%o (Haskell)

%o a051178 n = a051178_list !! (n-1)

%o a051178_list = filter (\x -> a027423 x `mod` x == 0) [1..]

%o -- _Reinhard Zumkeller_, Feb 27 2013

%o (PARI) valp(n,p)=my(s); while(n\=p, s+=n); s

%o 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

%Y Cf. A000005, A000142, A027423, A277166.

%K nonn,nice

%O 1,2

%A _Leroy Quet_