A049645 Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.

1, 3, 7, 11, 19, 21, 23, 31, 33, 35, 43, 47, 57, 59, 62, 67, 69, 71, 77, 79, 83, 91, 93, 94, 103, 105, 107, 115, 119, 127, 129, 131, 133, 139, 141, 151, 155, 158, 161, 163, 167, 177, 179, 186, 189, 191, 199, 201, 203, 209, 211, 213, 217, 223

Offset

1,2

Comments

Bateman et al. (1981) proved that the asymptotic density of this sequence is 1/2. - Amiram Eldar, Jan 16 2020

References

Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, section 51, page 119.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, The arithmetic mean of the divisors of an integer, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, alternative link.

Formula

{n: A035116(n) | A000203(n)}. - R. J. Mathar, Jan 29 2019

Maple

with(numtheory): t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod sigma[ 0 ](n)^2 = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t;

Mathematica

Select[Range[1, 250], Mod[DivisorSigma[1, #], DivisorSigma[0, #]^2] == 0 &] (* G. C. Greubel, Dec 06 2017 *)

Prog

(PARI) isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ Michel Marcus, Dec 07 2017
(Magma) [k:k in [1..230]| DivisorSigma(1, k) mod (DivisorSigma(0, k))^2 eq 0]; // Marius A. Burtea, Jan 16 2020

Crossrefs

Cf. A003601, A049642, A049692.

Cf. A000203, A035116.

Sequence in context: A097748 A255000 A244570 * A167181 A049835 A117510

Adjacent sequences: A049642 A049643 A049644 * A049646 A049647 A049648

Keyword

nonn

Author

N. J. A. Sloane

Status

approved