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

1, 4, 36, 576, 14400, 705600, 57153600, 6915585600, 1168733966400, 337764116289600, 121932845980545600, 64502475523708622400, 40314047202317889000000, 33904113697149344649000000, 32581853262960520207689000000, 44604557116992952164326241000000, 74980260513665152588232411121000000

Offset: 1

Author

Amiram Eldar, May 01 2019

Keywords

• nonn

Comments

Subsequence of A025487.

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, ...

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..201

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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