A034683 - OEIS

%I #39 Mar 12 2026 06:56:26

%S 30,42,66,70,78,102,114,138,150,174,186,210,222,246,258,282,294,318,

%T 330,354,366,390,402,420,426,438,462,474,498,510,534,546,570,582,606,

%U 618,630,642,654,660,678,690,714,726,750,762,770,780,786,798,822,834

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

%C 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

%C 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

%C The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - _Amiram Eldar_, Apr 18 2024

%C 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

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

%H Nathaniel Johnston, <a href="/A034683/b034683.txt">Table of n, a(n) for n = 1..10000</a>

%H Charles Robert Wall, <a href="https://trace.tennessee.edu/utk_graddiss/8570/">Topics related to the sum of unitary divisors of an integer</a>, Ph.D. diss., University of Tennessee, 1970.

%p isA034683 := proc(n)

%p     is(A034448(n) > 2*n) ;

%p end proc:

%p for n from 1 do

%p     if isA034683(n) then

%p         print(n);

%p     end if;

%p end do: # _R. J. Mathar_, Nov 10 2014

%t usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];

%t Select[Range[1000], usigma[#] > 2#&] (* _Jean-François Alcover_, Mar 23 2020, after _Giovanni Resta_ in A034448 *)

%o (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

%Y Subsequence of A005101.

%Y Cf. A034444, A034448, A129485 (odd terms), A129487.

%Y See the comments for the relationships with A001694, A052485, A061742, A082020, A302993,

%K nonn

%O 1,1

%A _Erich Friedman_