cp's OEIS Frontend

A318278 Exponential highly composite numbers: where the number of exponential divisors of n (A049419) increases to a record.

%I A318278 #19 Feb 16 2025 08:33:56
%S A318278 1,4,16,36,144,576,1296,3600,14400,32400,129600,705600,1587600,
%T A318278 6350400,39690000,57153600,158760000,768398400,4802490000,6915585600,
%U A318278 19209960000,129859329600,811620810000,1168733966400,3246483240000,29218349160000,159077678760000
%N A318278 Exponential highly composite numbers: where the number of exponential divisors of n (A049419) increases to a record.
%C A318278 Analogous to highly composite numbers (A002182) with number of exponential divisors (A049419) instead of number of divisors (A000005). The record numbers of exponential divisors are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 192, 216, 256, 288, 384, 432, ... The numbers have each exponent > 1. Proof: Suppose for some m, the exponent of prime p is 1. Then m/p has the same number of prime divisors hence m isn't a record. A contradiction hence all exponents are > 1.
%C A318278 Terms have even exponents in A025487 in their prime factorization.
%C A318278 Proof: Suppose some exponent e of a prime is not in A025487. Then there exists some term in A025487 number e1 < e that has the same number of divisors since e > 1. A contradiction hence all exponents are in A025487 and > 1.
%C A318278 By the above discussion, the terms are squares with their square roots: 1,2,4,6,12,24,36,60,120,180,360,840,1260,2520,6300.
%C A318278 The above argument can be trivially modified to further restrict the possible exponents to members of A002182: replace "the same number of" with "at least as many". - _Charlie Neder_, Oct 27 2018
%H A318278 Charlie Neder, <a href="/A318278/b318278.txt">Table of n, a(n) for n = 1..180</a>
%H A318278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>
%e A318278 144 is in the sequence since it has 6 exponential divisors (being 6, 12, 18, 36, 48, 144), and no positive integer < 144 has at least 6 exponential divisors hence 144 is in the sequence.
%t A318278 edivnum[1] = 1; edivnum [p_?PrimeQ] = 1; edivnum [p_?PrimeQ, e_] := DivisorSigma[ 0, e ]; edivnum [n_] := Times @@ (edivnum [#[[1]], #[[2]]] & ) /@ FactorInteger[ n ];  em = 0; s = {}; Do[e =edivnum [k]; If[e >em, AppendTo[s, k]; em = e], {k, 1, 100000}]; s (* after Jean-François Alcover in A049419 *)
%Y A318278 Cf. A002182, A025487, A049419, A293185.
%K A318278 nonn
%O A318278 1,2
%A A318278 _Amiram Eldar_ and _David A. Corneth_, Aug 29 2018