cp's OEIS Frontend

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

%I A049645 #24 Sep 08 2022 08:44:58
%S A049645 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 A049645 103,105,107,115,119,127,129,131,133,139,141,151,155,158,161,163,167,
%U A049645 177,179,186,189,191,199,201,203,209,211,213,217,223
%N A049645 Numbers k such that the square of the number of divisors of k divides the sum of the divisors of k.
%C A049645 Bateman et al. (1981) proved that the asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Jan 16 2020
%D A049645 Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.
%D A049645 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, section 51, page 119.
%H A049645 G. C. Greubel, <a href="/A049645/b049645.txt">Table of n, a(n) for n = 1..5000</a>
%H A049645 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 A049645 {n: A035116(n) | A000203(n)}. - _R. J. Mathar_, Jan 29 2019
%p A049645 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 A049645 Select[Range[1, 250], Mod[DivisorSigma[1, #], DivisorSigma[0, #]^2] == 0 &] (* _G. C. Greubel_, Dec 06 2017 *)
%o A049645 (PARI) isok(n) = sigma(n) % numdiv(n)^2 == 0; \\ _Michel Marcus_, Dec 07 2017
%o A049645 (Magma) [k:k in [1..230]| DivisorSigma(1,k) mod (DivisorSigma(0,k))^2 eq 0]; // _Marius A. Burtea_, Jan 16 2020
%Y A049645 Cf. A003601, A049642, A049692.
%Y A049645 Cf. A000203, A035116.
%K A049645 nonn
%O A049645 1,2
%A A049645 _N. J. A. Sloane_