A307870 Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.

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

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 k-th 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: A005934 A085629 A349111 * A330873 A233442 A046055

Adjacent sequences: A307867 A307868 A307869 * A307871 A307872 A307873

Keyword

nonn

Author

Amiram Eldar, May 02 2019

Status

approved