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A098109 a(n) is the least number k such that the number of divisors of k! exceeds 10^n.

%I A098109 #13 Feb 03 2025 02:15:33
%S A098109 5,9,13,17,23,29,34,40,46,53,59,67,73,79,87,95,103,109,116,127,134,
%T A098109 141,150,158,167,175,182,193,199,210,218,227,234,242,254,263,271,281,
%U A098109 290,301,311,317,329,337,349,358,367,379,387,397,406,418,427,436,446,455
%N A098109 a(n) is the least number k such that the number of divisors of k! exceeds 10^n.
%H A098109 Amiram Eldar, <a href="/A098109/b098109.txt">Table of n, a(n) for n = 1..10000</a>
%p A098109 # multiply two ifactor representations [p1,e1],[p2,e2],[p3,e2]
%p A098109 mulif := proc(if1, if2)
%p A098109     local ifr,t,p,e,ix,ifi ;
%p A098109     ifr := if1 ;
%p A098109     for t in if2 do
%p A098109         p := op(1,t) ;
%p A098109         e := op(2,t) ;
%p A098109         ix := 0 ;
%p A098109         for ifi from 1 to nops(ifr) do
%p A098109             if op(1,op(ifi,ifr)) = p then
%p A098109                 ix := ifi;
%p A098109                 break;
%p A098109             end if;
%p A098109         end do:
%p A098109         if ix = 0 then
%p A098109             ifr := [op(ifr),[p,e]] ;
%p A098109         else
%p A098109             e := e+op(2,op(ix,ifr)) ;
%p A098109             ifr := subsop(ix=[p,e],ifr) ;
%p A098109         end if;
%p A098109     end do:
%p A098109     return ifr ;
%p A098109 end proc:
%p A098109 # tau(iff) using multiplicative property of tau
%p A098109 tauif := proc(iff)
%p A098109     local r;
%p A098109     r := 1 ;
%p A098109     for t in iff do
%p A098109         r := r*(1+op(2,t)) ;
%p A098109     end do:
%p A098109     return r;
%p A098109 end proc:
%p A098109 # ifactor representation of m!
%p A098109 iffact := proc(m)
%p A098109     local r,f ;
%p A098109     if m <=1 then
%p A098109         return [] ;
%p A098109     else
%p A098109         r := [[2,1]] ;
%p A098109         for f from 3 to m do
%p A098109             ifmf := ifactors(f)[2] ;
%p A098109             r := mulif(r,ifmf) ;
%p A098109         end do:
%p A098109         return r;
%p A098109     end if:
%p A098109 end proc:
%p A098109 A027423 := proc(n)
%p A098109     iffact(n) ;
%p A098109     tauif(%) ;
%p A098109 end proc:
%p A098109 A098109 := proc(n)
%p A098109     local m ;
%p A098109     for m from 2 do
%p A098109         if A027423(m) > 10^n then
%p A098109             return m;
%p A098109         end if;
%p A098109     end do:
%p A098109 end proc:
%p A098109 for n from 1 do
%p A098109     print(A098109(n)) ;
%p A098109 end do: # _R. J. Mathar_, Nov 19 2011
%o A098109 (PARI) A027423(n) = {my(prd = 1); forprime(p = 2, n, prd *= (1 + (n - sumdigits(n, p))/(p-1))); prd;}
%o A098109 list(lim) = {my(pow = 10); for(k = 1, lim, if(A027423(k) > pow, print1(k, ", "); pow * = 10));} \\ _Amiram Eldar_, Feb 03 2025
%Y A098109 Cf. A027423.
%K A098109 nonn
%O A098109 1,1
%A A098109 _Jeff Burch_, Sep 23 2004