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A092593 a(n) is the smallest number k > 1 for which A001142(k)/A002944(k+1)^n is an integer.

%I A092593 #19 Jan 24 2019 10:00:11
%S A092593 2,3,9,9,15,15,38,45,45,45,61,61,225,225,225,225,225,225,225,225,225,
%T A092593 225,635,635,1545,1545,1545,1545,2137,2137,2137,2137,2137,2137,2137,
%U A092593 2137,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660,2660
%N A092593 a(n) is the smallest number k > 1 for which A001142(k)/A002944(k+1)^n is an integer.
%C A092593 a(62) > 12500. - _Robert Israel_, Jan 24 2019
%e A092593 n=4, A001142(9) = 1*9*36*...*9*1 = 11759522374656,
%e A092593 A002944(10) = lcm(1,2,...,10)/10=252 and A001142(9) = 2916*(252^4) = 11759522374656,
%e A092593 so a(4)=9, the smallest relevant number.
%p A092593 A001142:= proc(n) option remember; procname(n-1)*n^(n-1)/(n-1)! end proc:
%p A092593 A001142(0):= 1:
%p A092593 A002944:= proc(n) option remember; ilcm(n,procname(n-1)*(n-1))/n end proc:
%p A092593 A002944(1):= 1:
%p A092593 f:= proc(n) option remember; local k;
%p A092593 for k from procname(n-1) do
%p A092593    if type(A001142(k)/A002944(k+1)^n, integer) then return k fi
%p A092593 od
%p A092593 end proc:
%p A092593 f(1):= 2:
%p A092593 map(f, [$1..61]); # _Robert Israel_, Jan 23 2019
%t A092593 Table[fla=1;Do[s1=Apply[Times, Table[Binomial[n, j], {j, 0, n}]]; s2=Apply[LCM, Table[Binomial[n, j], {j, 0, n}]]; If[IntegerQ[s1/(s2^k)]&&!Equal[n, 1]&&Equal[fla, 1], Print[{n, k}];fla=0], {n, 1, 230}], {k, 1, 25}]
%Y A092593 Cf. A001142, A002944, A092593.
%K A092593 nonn
%O A092593 1,1
%A A092593 _Labos Elemer_, Mar 10 2004
%E A092593 Corrected and extended by _Robert Israel_, Jan 23 2019