

A307870


Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.


3



1, 4, 8, 16, 32, 64, 128, 256, 432, 576, 864, 1296, 1728, 2592, 3456, 5184, 6912, 10368, 15552, 20736, 31104, 41472, 62208, 82944, 93312, 124416, 186624, 248832, 373248, 497664, 746496, 995328, 1119744, 1492992, 2239488, 2592000, 2985984, 3888000, 5184000, 7776000
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OFFSET

1,2


COMMENTS

Numbers k with d(k)/2^omega(k) > d(j)/2^omega(j) for all j < k, where d(k) is the number of divisors of k (A000005), and omega(k) is the number of distinct prime factors of k (A001221), so 2^omega(k) is the number of unitary divisors of k (A034444).
Subsequence of A025487.
The first term that is divisible by the kth prime is 4, 432, 2592000, 53343360000, 134190022982400000, 35377857659079936000000, 160601747163451186424832000000, 35800939973308629849857487360000000, ...
All the terms are powerful (A001694), since if p is a prime factor of k with multuplicity 1, then k and k/p have the same ratio.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..809


EXAMPLE

All squarefree numbers k have d(k)/ud(k) = 1. Thus 4, the first nonsquarefree number, has a record value of d(4)/ud(4) = 3/2 and thus it is in the sequence.


MATHEMATICA

r[n_] := DivisorSigma[0, n]/(2^PrimeNu[n]); rm = 0; n = 1; s = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[s, n]]; n++, {10^7}]; s


CROSSREFS

Cf. A000005, A001221, A001694, A025487, A034444, A285906, A307869.
Sequence in context: A298807 A005934 A085629 * A233442 A046055 A186949
Adjacent sequences: A307867 A307868 A307869 * A307871 A307872 A307873


KEYWORD

nonn


AUTHOR

Amiram Eldar, May 02 2019


STATUS

approved



