A218976 a(n) is the smallest positive integer such that 10^(2 + floor(k/a(1)) + floor(k/a(2)) + ... + floor(k/a(n))) divides (k+9)! for all k > 0.

6, 16, 116, 241, 242, 491, 991, 2491, 3331, 14966, 15556, 62491, 78116, 83331, 249991, 264866, 546841, 1109366, 2265491, 4999861, 4999991, 5837041, 12499996, 25249861, 26011861, 36249091, 80070866, 190999991, 242090611, 365038621, 976562241, 1210466866, 1830622801

Offset: 1

Author

R. J. Cano and Charles R Greathouse IV, Nov 08 2012

Keywords

• nonn

Comments

Every factorial of the form (k+9)! for every integer k > 0 ends at least in two zeros. This sequence gives a lower bound on the number of zeros.

This sequence is infinite and increasing, with 1/a(1) + 1/a(2) + ... = 1/4.

Conjecture: All terms except a(5) are 1 mod 5. - R. J. Cano, Nov 11 2012

Examples

			10^(2 + floor(5/5)) does not divide 14!, so a(1) > 5. But 10^(2 + floor(k/6)) divides (k+9)! for all k > 0, so a(1) = 6.
		

Links

• Table of n, a(n) for n = 1..33
• Dickson, Leonard Eugene (2005) [1919], History of the theory of numbers. Vol. I: Divisibility and primality., New York: Dover Publications, ISBN 978-0-486-44232-7, [MR], page 263. De Polignac's formula
• Wikipedia, De Polignac's formula
• Wikipedia, Trailing zero

Crossrefs

Cf. A027868, A217626.

Programs

PARI

searchLimit(s1)={
    my(e,s2,f=(e,s)->(e+2-9*s)/(s-s1));
    while(s2<=s1, s2 += 1/5^e++);
    min(f(e,s2), f(e++,s2+=1/5^e))\1
};
v5(n)=my(s);while(n\=5,s+=n);s;
nxt(v=[6])={
    my(S=sum(i=1,#v,1/v[i]), candidate=max(v[#v],1\(1/4-S))+1, k=candidate, lm=searchLimit(S+1/candidate));
    while(k<=lm,
        if(v5(k+9)<2+sum(i=1,#v,k\v[i])+k\candidate,
            candidate++;
            lm=searchLimit(S+1/candidate)
        ,
            k++
        )
    );
    candidate
};
steps(n)={
    my(v=[6],t);
    print1(6);
    for(i=2,n,
        t=nxt(v);
        print1(", "t);
        v=concat(v,t)
    );
    v
};
steps(20)

Formula

Let Psi(k) = 2 + sum_{n >= 1} floor(k/a(n)). Then 10^Psi(k) divides (k+9)!.

Extensions

a(32)-a(33) from Charles R Greathouse IV, Nov 19 2012