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Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.
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%I #24 Sep 08 2022 08:44:58

%S 1,3,7,11,19,21,23,31,33,35,43,47,57,59,62,67,69,71,77,79,83,91,93,94,

%T 103,105,107,115,119,127,129,131,133,139,141,151,155,158,161,163,167,

%U 177,179,186,189,191,199,201,203,209,211,213,217,223

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

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

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

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

%H G. C. Greubel, <a href="/A049645/b049645.txt">Table of n, a(n) for n = 1..5000</a>

%H Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, <a href="https://doi.org/10.1007/BFb0096462">The arithmetic mean of the divisors of an integer</a>, 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, <a href="https://math.dartmouth.edu/~carlp/PDF/31.pdf">alternative link</a>.

%F {n: A035116(n) | A000203(n)}. - _R. J. Mathar_, Jan 29 2019

%p 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;

%t Select[Range[1, 250], Mod[DivisorSigma[1, #], DivisorSigma[0, #]^2] == 0 &] (* _G. C. Greubel_, Dec 06 2017 *)

%o (PARI) isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ _Michel Marcus_, Dec 07 2017

%o (Magma) [k:k in [1..230]| DivisorSigma(1,k) mod (DivisorSigma(0,k))^2 eq 0]; // _Marius A. Burtea_, Jan 16 2020

%Y Cf. A003601, A049642, A049692.

%Y Cf. A000203, A035116.

%K nonn

%O 1,2

%A _N. J. A. Sloane_