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A306736 Exponential infinitary highly composite numbers: where the number of exponential infinitary divisors (A307848) increases to record.

%I A306736 #29 May 04 2019 16:05:31
%S A306736 1,4,36,576,14400,705600,57153600,6915585600,1168733966400,
%T A306736 337764116289600,121932845980545600,64502475523708622400,
%U A306736 40314047202317889000000,33904113697149344649000000,32581853262960520207689000000,44604557116992952164326241000000,74980260513665152588232411121000000
%N A306736 Exponential infinitary highly composite numbers: where the number of exponential infinitary divisors (A307848) increases to record.
%C A306736 Subsequence of A025487.
%C A306736 All the terms have prime factors with multiplicities which are infinitary highly composite number (A037992) > 1, similarly to exponential highly composite numbers (A318278) whose prime factors have multiplicities which are highly composite numbers (A002182). Thus all the terms are squares. Their square roots are 1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, ...
%C A306736 Differs from A307845 (exponential unitary highly composite numbers) from n >= 107. a(107) = 2^24 * (3 * 5 * ... * 19)^6 * (23 * 29 * ... * 509)^2 ~ 2.370804... * 10^456, while A307845(107) = (2 * 3 * 5 * ... * 19)^6 * (23 * 29 * ... * 521)^2 ~ 2.454885... * 10^456.
%H A306736 Amiram Eldar, <a href="/A306736/b306736.txt">Table of n, a(n) for n = 1..201</a>
%F A306736 A307848(a(n)) = 2^(n-1).
%t A306736 di[1] = 1; di[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[All, 2]]]; fun[p_, e_] := di[e]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); s = {}; am = 0; Do[a1 = a[n]; If[a1 > am, am = a1; AppendTo[s, n]], {n, 1, 10^6}]; s (* after _Jean-François Alcover_ at A037445 *)
%Y A306736 Cf. A002182, A025487, A037445, A037992, A307845, A307848, A318278.
%K A306736 nonn
%O A306736 1,2
%A A306736 _Amiram Eldar_, May 01 2019