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A173615 Numbers k such that rad(k)^2 divides sigma(k).

%I A173615 #32 Jan 30 2025 05:09:24
%S A173615 1,96,864,1080,1782,6144,7128,7776,17280,27000,28512,54432,55296,
%T A173615 69984,87480,114048,215622,276480,381024,393216,432000,433026,456192,
%U A173615 497664,629856,675000,862488,1382400,1399680,1677312,1732104,1824768,2187000,2195424,2667168
%N A173615 Numbers k such that rad(k)^2 divides sigma(k).
%C A173615 rad(n) is the product of the primes dividing n (A007947) and sigma(n) = sum of divisors of n (A000203). Considering the integers k = (2^a)*(3^b), where a+1 = 6*m and b >= 1, we obtain an infinite number of numbers such that rad(k)^2 divides sigma(k).
%C A173615 De Koninck (2000, 2002) asked whether 1 and 1782 are the only numbers k such that rad(k)^2 = sigma(k). - _Amiram Eldar_, Jan 29 2025
%D A173615 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A173615 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 102.
%H A173615 Amiram Eldar, <a href="/A173615/b173615.txt">Table of n, a(n) for n = 1..500</a> (terms 1..300 from Donovan Johnson)
%H A173615 Kevin A. Broughan, Jean-Marie De Koninck, Imre Kátai, and Florian Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Broughan/broughan20.html">On integers for which the sum of divisors is the square of the squarefree core</a>, J. Integer Seq., 15 (2012), Article 12.7.5, pp. 1-12. See Final remarks pp. 10-11.
%H A173615 Kevin Broughan Daniel Delbourgo, and Qizhi Zhou, <a href="https://doi.org/10.1016/j.jnt.2013.10.011">A conjecture of De Koninck regarding particular square values of the sum of divisors function</a>, Journal of Number Theory, Vol. 137 (2014), pp. 50-66.
%H A173615 Yong-Gao Chen and Xin Tong, <a href="https://doi.org/10.1016/j.jnt.2015.02.004">On a conjecture of de Koninck</a>, Journal of Number Theory, Vol. 154 (2015), pp. 324-364.
%H A173615 Jean-Marie De Koninck, <a href="https://westcoastnumbertheory.org/wp-content/uploads/2012/09/wcnt-problems-2000.pdf">Probelm 000:08</a>, Western Number Theory Problems, 17 & 20 Dec 2000, edited by Gerry Myerson, p. 5.
%H A173615 Jean-Marie De Koninck, <a href="https://doi.org/10.2307/3072407">Problem 10966</a>, The American Mathematical Monthly, Vol. 109, No. 8 (2002), p. 759; <a href="https://doi.org/10.2307/4145084">Editorial comment</a>, ibid., Vol. 111, No. 6 (2004), p. 536.
%H A173615 Min Tang and Zhi-Jun Zhou, <a href="https://math.colgate.edu/~integers/s60/s60.Abstract.html">On a conjecture of De Koninck</a>, INTEGERS, Vol. 18 (2018), Article #A60.
%H A173615 Wacław Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Elementary theory of numbers, Warszawa, 1964.
%H A173615 Tomohiro Yamada, <a href="https://doi.org/10.2140/moscow.2021.10.249">On a problem of De Koninck</a>, Moscow Journal of Combinatorics and Number Theory, Vol. 10, No. 3 (2021), pp. 249-260; <a href="http://dx.doi.org/10.2140/moscow.2021.10.339">Correction</a>, ibid., Vol. 10, No. 4 (2021), p. 339; <a href="https://arxiv.org/abs/1906.10001">arXiv preprint</a>, arXiv:1906.10001 [math.NT], 2019-2021.
%e A173615 rad(96)^2 = 6^2 = 36, sigma(96) = 252 and 36 divides 252.
%p A173615 for n from 1 to 2000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if irem(sigma(n),t2^2) = 0 then print (n): else fi: od :
%t A173615 f[p_, e_] := (p^(e+1) - 1)/(p^2 * (p-1)); q[k_] := IntegerQ[Times @@ f @@@ FactorInteger[k]]; q[1] = True; Select[Range[3*10^6], q] (* _Amiram Eldar_, Jan 29 2025 *)
%o A173615 (PARI) isok(n) = my(f=factor(n)); (sigma(f) % factorback(f[, 1])^2) == 0; \\ _Michel Marcus_, Nov 09 2020
%Y A173615 Cf. A000203, A007947, A078615.
%Y A173615 Subsequence of A175200.
%K A173615 nonn
%O A173615 1,2
%A A173615 _Michel Lagneau_, Feb 22 2010
%E A173615 a(30)-a(35) from _Donovan Johnson_, Jan 14 2012