A307112 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.

180, 420, 504, 660, 780, 1584, 1848, 1872, 1890, 2184, 2352, 2376, 2772, 2856, 3150, 3192, 3276, 4284, 4410, 4788, 4896, 5100, 5292, 5700, 5796, 6864, 6900, 6930, 7344, 7728, 8190, 8208, 8424, 9744, 10296, 10416, 10710, 10944, 11550, 11970, 12012, 12432, 12870

Offset: 1

Author

Amiram Eldar, Mar 25 2019

Keywords

• nonn

Comments

Analogous to A071395 with abundancy index 3 instead of 2.

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Graeme L. Cohen, Primitive alpha-abundant numbers, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.
• Paul Erdős, On additive arithmetical functions and applications of probability to number theory, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3 (1956), pp. 13-19.
• Paul Erdős, Remarks on number theory. I: On primitive alpha-abundant numbers, Acta Arithmetica., Vol. 5, No. 1 (1959), pp. 25-33, alternative link.

Crossrefs

Cf. A000203, A068403, A071395.

Programs

Mathematica

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