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

5, 9, 13, 17, 23, 29, 34, 40, 46, 53, 59, 67, 73, 79, 87, 95, 103, 109, 116, 127, 134, 141, 150, 158, 167, 175, 182, 193, 199, 210, 218, 227, 234, 242, 254, 263, 271, 281, 290, 301, 311, 317, 329, 337, 349, 358, 367, 379, 387, 397, 406, 418, 427, 436, 446, 455

Offset: 1

Author

Jeff Burch, Sep 23 2004

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A027423.

Programs

Maple

# multiply two ifactor representations [p1,e1],[p2,e2],[p3,e2]
mulif := proc(if1, if2)
    local ifr,t,p,e,ix,ifi ;
    ifr := if1 ;
    for t in if2 do
        p := op(1,t) ;
        e := op(2,t) ;
        ix := 0 ;
        for ifi from 1 to nops(ifr) do
            if op(1,op(ifi,ifr)) = p then
                ix := ifi;
                break;
            end if;
        end do:
        if ix = 0 then
            ifr := [op(ifr),[p,e]] ;
        else
            e := e+op(2,op(ix,ifr)) ;
            ifr := subsop(ix=[p,e],ifr) ;
        end if;
    end do:
    return ifr ;
end proc:
# tau(iff) using multiplicative property of tau
tauif := proc(iff)
    local r;
    r := 1 ;
    for t in iff do
        r := r*(1+op(2,t)) ;
    end do:
    return r;
end proc:
# ifactor representation of m!
iffact := proc(m)
    local r,f ;
    if m <=1 then
        return [] ;
    else
        r := [[2,1]] ;
        for f from 3 to m do
            ifmf := ifactors(f)[2] ;
            r := mulif(r,ifmf) ;
        end do:
        return r;
    end if:
end proc:
A027423 := proc(n)
    iffact(n) ;
    tauif(%) ;
end proc:
A098109 := proc(n)
    local m ;
    for m from 2 do
        if A027423(m) > 10^n then
            return m;
        end if;
    end do:
end proc:
for n from 1 do
    print(A098109(n)) ;
end do: # R. J. Mathar, Nov 19 2011

PARI

A027423(n) = {my(prd = 1); forprime(p = 2, n, prd *= (1 + (n - sumdigits(n, p))/(p-1))); prd;}
list(lim) = {my(pow = 10); for(k = 1, lim, if(A027423(k) > pow, print1(k, ", "); pow * = 10));} \\ Amiram Eldar, Feb 03 2025