A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834

Offset

1,1

Comments

If a term n in the sequence ends in neither 0 nor 5, then 10*n is also in the sequence. - Lekraj Beedassy, Jun 11 2004

The lower asymptotic density of this sequence is larger than 1/18 = 0.0555... which is the density of its subsequence of numbers of the form 6*m where gcd(m, 6) = 1 and m > 1. Numerically, based on counts of terms below 10^n (A302993), it seems that this sequence has an asymptotic density which equals to about 0.070034... - Amiram Eldar, Feb 13 2021

The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - Amiram Eldar, Apr 18 2024

All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, usigma(k)/k < 15/Pi^2 = 1.519817... (A082020; the record values are attained at the squares of primorials, A061742). - Amiram Eldar, Jul 20 2024

References

C. Sung, Mathematical Buds, "Unitary Divisors", Chap. V, pp. 42-67, Ed. H. D. Ruderman, Mu Alpha Theta OK 1978.

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Maple

isA034683 := proc(n)
    is(A034448(n) > 2*n) ;
end proc:
for n from 1 do
    if isA034683(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

Mathematica

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
Select[Range[1000], usigma[#] > 2#&] (* Jean-François Alcover, Mar 23 2020, after Giovanni Resta in A034448 *)

Prog

(PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]) > 2*n; } \\ Amiram Eldar, Apr 18 2024

Crossrefs

Subsequence of A005101.

Cf. A034444, A034448, A129487, A302993.

Cf. A001694, A061742, A082020.

Sequence in context: A053858 A075819 A306217 * A328328 A302574 A087248

Adjacent sequences: A034680 A034681 A034682 * A034684 A034685 A034686

Keyword

nonn

Author

Erich Friedman

Status

approved