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Primitive 3-abundant numbers: Numbers k such that sigma(k) > 3k (A068403) all of whose proper divisors d are 3-deficient numbers having sigma(d) < 3d.
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%I #12 Mar 25 2019 19:24:06

%S 180,420,504,660,780,1584,1848,1872,1890,2184,2352,2376,2772,2856,

%T 3150,3192,3276,4284,4410,4788,4896,5100,5292,5700,5796,6864,6900,

%U 6930,7344,7728,8190,8208,8424,9744,10296,10416,10710,10944,11550,11970,12012,12432,12870

%N Primitive 3-abundant numbers: Numbers k such that sigma(k) > 3k (A068403) all of whose proper divisors d are 3-deficient numbers having sigma(d) < 3d.

%C Analogous to A071395 with abundancy index 3 instead of 2.

%D Paul Erdős and János Surányi, Topics in the Theory of Numbers, New York: Springer, 2003, p. 243.

%H Amiram Eldar, <a href="/A307112/b307112.txt">Table of n, a(n) for n = 1..10000</a>

%H Graeme L. Cohen, <a href="https://doi.org/10.1090/S0025-5718-1984-0744936-X ">Primitive alpha-abundant numbers</a>, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.

%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1956-08.pdf">On additive arithmetical functions and applications of probability to number theory</a>, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3 (1956), pp. 13-19.

%H Paul Erdős, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav5i1p25bwm">Remarks on number theory. I: On primitive alpha-abundant numbers</a>, Acta Arithmetica., Vol. 5, No. 1 (1959), pp. 25-33, <a href="https://users.renyi.hu/~p_erdos/1959-20.pdf">alternative link</a>.

%t Select[Range@50000, DivisorSigma[1, #] > 3 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 3 # &, Most@ Divisors@ #] == 1 &] (* after _Michael De Vlieger_ at A071395 *)

%Y Cf. A000203, A068403, A071395.

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 25 2019