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

1,2

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A139033 A370753 A001044 * A307845 A346292 A086879

Adjacent sequences: A306733 A306734 A306735 * A306737 A306738 A306739

Keyword

nonn

Author

Amiram Eldar, May 01 2019

Status

approved