A068306 - OEIS

A068306

Numbers n such that n*tau(n) < sigma(n)*bigomega(n) where tau(n)=A000005(n), sigma(n)=A000203(n), and bigomega(n)=A001222(n).

4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 486, 512, 544, 576, 608, 640, 704, 729, 736, 768, 832

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence contains numbers of the form p^u, u >= p, p prime; numbers of the form 2^k*p, p odd prime and k >= floor(log(p)/log(2)). Example: the first number of the form 2^k*67 in the sequence is 2^6*67=4288 because floor(log(67)/log(2))=6.

[Are we to understand that these are all the numbers in the sequence, or just some of them? - N. J. A. Sloane, Dec 27 2018]

If n=p^(p-1) with p prime, n*tau(n) - sigma(n)*bigomega(n) = 1.

From David A. Corneth, Aug 24 2020: (Start)

If n > 1 then each of n, tau(n), sigma(n) and bigomega(n) are > 0. We can then write the inequality as tau(n)/bigomega(n) < sigma(n)/n.

Note that tau(n) and bigomega(n) only depend on the prime signature of n.

Suppose we choose some k from A025487. If k is not a term then no number with that prime signature is not a term. k is the number with the value for sigma(m)/m for numbers m with the same prime signature as k. (End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4000 terms from Robert Israel)

MAPLE

filter:= proc(n) local L, t, j;

L:= ifactors(n)[2];

n * mul(t[2]+1, t=L) < mul(add(t[1]^j, j=0..t[2]), t=L)*add(t[2], t=L)

end proc:

select(filter, [$1..1000]); # Robert Israel, Dec 26 2018

MATHEMATICA

filterQ[n_] := n DivisorSigma[0, n] < DivisorSigma[1, n] PrimeOmega[n];

Select[Range[1000], filterQ] (* Jean-François Alcover, Aug 24 2020 *)

PROG