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

1, 4, 16, 36, 144, 576, 1296, 3600, 14400, 32400, 129600, 705600, 1587600, 6350400, 39690000, 57153600, 158760000, 768398400, 4802490000, 6915585600, 19209960000, 129859329600, 811620810000, 1168733966400, 3246483240000, 29218349160000, 159077678760000

Offset: 1

Author

Amiram Eldar and David A. Corneth, Aug 29 2018

Keywords

• nonn

Comments

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.

Terms have even exponents in A025487 in their prime factorization.

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.

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.

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

Examples

			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.
		

Links

• Charlie Neder, Table of n, a(n) for n = 1..180
• Eric Weisstein's World of Mathematics, e-Divisor

Crossrefs

Cf. A002182, A025487, A049419, A293185.

Programs

Mathematica

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 *)