A306736 - OEIS

%I #29 May 04 2019 16:05:31

%S 1,4,36,576,14400,705600,57153600,6915585600,1168733966400,

%T 337764116289600,121932845980545600,64502475523708622400,

%U 40314047202317889000000,33904113697149344649000000,32581853262960520207689000000,44604557116992952164326241000000,74980260513665152588232411121000000

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

%C Subsequence of A025487.

%C 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 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 Amiram Eldar, <a href="/A306736/b306736.txt">Table of n, a(n) for n = 1..201</a>

%F A307848(a(n)) = 2^(n-1).

%t 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 Cf. A002182, A025487, A037445, A037992, A307845, A307848, A318278.

%K nonn

%O 1,2

%A _Amiram Eldar_, May 01 2019