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%I #61 Apr 17 2016 07:19:47
%S 6,16,116,241,242,491,991,2491,3331,14966,15556,62491,78116,83331,
%T 249991,264866,546841,1109366,2265491,4999861,4999991,5837041,
%U 12499996,25249861,26011861,36249091,80070866,190999991,242090611,365038621,976562241,1210466866,1830622801
%N 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.
%C 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.
%C This sequence is infinite and increasing, with 1/a(1) + 1/a(2) + ... = 1/4.
%C Conjecture: All terms except a(5) are 1 mod 5. - _R. J. Cano_, Nov 11 2012
%H <a href="/A218976/b218976.txt">Table of n, a(n) for n = 1..33</a>
%H 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, <a href="http://ams.org/mathscinet-getitem?mr=0245499">[MR]</a>, page 263. De Polignac's formula
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/De_Polignac%27s_formula">De Polignac's formula</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Trailing_zero">Trailing zero</a>
%F Let Psi(k) = 2 + sum_{n >= 1} floor(k/a(n)). Then 10^Psi(k) divides (k+9)!.
%e 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.
%o (PARI) searchLimit(s1)={
%o my(e,s2,f=(e,s)->(e+2-9*s)/(s-s1));
%o while(s2<=s1, s2 += 1/5^e++);
%o min(f(e,s2), f(e++,s2+=1/5^e))\1
%o };
%o v5(n)=my(s);while(n\=5,s+=n);s;
%o nxt(v=[6])={
%o 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));
%o while(k<=lm,
%o if(v5(k+9)<2+sum(i=1,#v,k\v[i])+k\candidate,
%o candidate++;
%o lm=searchLimit(S+1/candidate)
%o ,
%o k++
%o )
%o );
%o candidate
%o };
%o steps(n)={
%o my(v=[6],t);
%o print1(6);
%o for(i=2,n,
%o t=nxt(v);
%o print1(", "t);
%o v=concat(v,t)
%o );
%o v
%o };
%o steps(20)
%Y Cf. A027868, A217626.
%K nonn
%O 1,1
%A _R. J. Cano_ and _Charles R Greathouse IV_, Nov 08 2012
%E a(32)-a(33) from _Charles R Greathouse IV_, Nov 19 2012
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