A307115 Primitive 5-abundant numbers: Numbers k such that sigma(k) > 5k (A215264) all of whose proper divisors d are 5-deficient numbers (having sigma(d) < 5d).

122522400, 147026880, 183783600, 205405200, 220540320, 232792560, 273873600, 328648320, 428828400, 492972480, 497296800, 514594080, 537213600, 563603040, 575134560, 581981400, 605404800, 627026400, 629909280, 670269600, 684684000, 710629920, 739458720, 745945200

Offset

1,1

Comments

Analogous to A071395 with abundancy index 5 instead of 2.

The term with the greatest abundancy is a(312) = 8172244080 = A307111(5). - Peter Munn, Jun 05 2026

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..1000

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.

Eric Weisstein's World of Mathematics, Abundancy.

Formula

omega(a(n)) = A001221(a(n)) >= 6. - David A. Corneth, Mar 26 2019

{a(n)} = {k >= 2 : A000203(k) > 5k and A000203(A395192(k)) < 5*A395192(k)}. - Peter Munn, Jun 05 2026

Mathematica

Select[Range@500000000, DivisorSigma[1, #] > 5 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 5 # &, Most@ Divisors@ #] == 1 &] (* after Michael De Vlieger at A071395 *)

Crossrefs

Cf. A000203, A001221, A071395, A215264, A307111, A395192.

Sequence in context: A175321 A380933 A215264 * A079332 A172749 A172781

Adjacent sequences: A307112 A307113 A307114 * A307116 A307117 A307118

Keyword

nonn

Author

Amiram Eldar, Mar 25 2019

Status

approved