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A049645
Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.
2
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
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
KEYWORD
nonn
STATUS
approved