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%I #40 Sep 06 2019 03:03:55
%S 5,13,19,32,73,89,140,199,294,468,1072,1072,1072,2161,2976,32805,
%T 32806,65732,65732,262153,262154,524457,524640,4194464,4194464,
%U 8388640,8388640,33554432,33554432,67108992,67109088,2147483659,2147484110,4294967312,4294967312,17179869209,17179869210
%N a(n) is the least number m such that (m+n)!/m! = (m+1)*(m+2)*...*(m+n) divides lcm(1,...,m).
%C From _David A. Corneth_, Aug 30 2019: (Start)
%C As (m+1)*(m+2)*...*(m+n) is the product of n consecutive integers, it's divisible by n! and so a(n) >= 2^A011371(n) = A060818(n).
%C None of m+1..m+n are prime. (End)
%H David A. Corneth, <a href="/A082093/b082093.txt">Table of n, a(n) for n = 1..75</a>
%e a(6)=89: lcm(1,...,89) = 718766754945489455304472257065075294400 is divisible by 625757605200 = 90*91*92*93*94*95 = (89+6)!/89! and the quotient is 1148634469597477063638686172.
%e For n=1 see A080765(1) = A082093(1).
%t k = 1; lc = 1; Do[While[lc = LCM[lc, k]; Mod[lc, (n + k)!/k! ] != 0, k++ ]; Print[{n, k}], {n, 0, 50}] (* _Robert G. Wilson v_, Apr 12 2006 *)
%Y Cf. A011371, A060818, A080765.
%K nonn
%O 1,1
%A _Labos Elemer_, Apr 10 2003
%E a(16)-a(19) from _Robert G. Wilson v_, Apr 12 2006
%E a(20)-a(23) from _Vaclav Kotesovec_, Aug 30 2019
%E a(24)-a(37) from _David A. Corneth_, Aug 30 2019
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