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

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A003601, A049642, A049692.

Cf. A000203, A035116.

Programs

Magma

[k:k in [1..230]| DivisorSigma(1,k) mod (DivisorSigma(0,k))^2 eq 0]; // Marius A. Burtea, Jan 16 2020

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 *)

PARI

isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ Michel Marcus, Dec 07 2017

Formula

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