%I #28 Oct 01 2014 11:43:31

%S 2,3,4,5,125,125,138,220,220,1766,5526,10351,12365,65653,65653,202738,

%T 490333,808762,1478432,1971352,1971352,1971352,14798206,14798206,

%U 14798206,14798206,161974053,547880880,1619543840,1619543840,1619543840,2103844465,6435961044

%N a(n) is the smallest k such that a unique product of distinct terms of A050376 which is equal to k! contains at least the first n terms of A050376.

%C By the definition, the representation of a(n)! as a product of distinct terms of A050376 should contain the first n terms of A050376 and there is no restriction on the distribution of other factors of this product.

%C a(38) > 2 * 10^11. - _Hiroaki Yamanouchi_, Oct 01 2014

%H Hiroaki Yamanouchi, <a href="/A240695/b240695.txt">Table of n, a(n) for n = 1..37</a>

%e 5! = 2*3*4*5. We have the first 4 terms of A050376, so a(4) = 5.

%t bad[n_, pp_, mo_] := Catch[Do[If[ Mod[(n - Total@ IntegerDigits[n, pp[[i]]]) /(pp[[i]] - 1), mo[[i]] + 1] != mo[[i]], Throw@ True], {i, Length@ pp}]; False]; a[n_]:= Block[{fa, mo, pp, k},fa = FactorInteger[ Times @@ Select[Range[2, Prime[n]], (f = FactorInteger@# ; Length[f] == 1 && IntegerQ[Log[2, f[[1, 2]]]]) &, n]]; pp = First /@ fa; mo = Last /@ fa; k = fa[[-1, 1]]; While[ bad[k, pp, mo], k++]; k]; Array[a,15] (* _Giovanni Resta_, Apr 11 2014 *)

%Y Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Apr 10 2014

%E a(5)-a(23) from _Giovanni Resta_, Apr 11 2014

%E a(24)-a(33) from _Hiroaki Yamanouchi_, Oct 01 2014