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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 (list; graph; refs; listen; history; text; internal format)
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: A298807 A005934 A085629 * A233442 A046055 A186949

Adjacent sequences:  A307867 A307868 A307869 * A307871 A307872 A307873

KEYWORD

nonn

AUTHOR

Amiram Eldar, May 02 2019

STATUS

approved

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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)