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

1, 96, 864, 1080, 1782, 6144, 7128, 7776, 17280, 27000, 28512, 54432, 55296, 69984, 87480, 114048, 215622, 276480, 381024, 393216, 432000, 433026, 456192, 497664, 629856, 675000, 862488, 1382400, 1399680, 1677312, 1732104, 1824768, 2187000, 2195424, 2667168

Offset: 1

Author

Michel Lagneau, Feb 22 2010

Keywords

• nonn

Comments

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).

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

Examples

			rad(96)^2 = 6^2 = 36, sigma(96) = 252 and 36 divides 252.
		

References

• 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.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 102.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..300 from Donovan Johnson)
• Kevin A. Broughan, Jean-Marie De Koninck, Imre Kátai, and Florian Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), Article 12.7.5, pp. 1-12. See Final remarks pp. 10-11.
• Kevin Broughan Daniel Delbourgo, and Qizhi Zhou, A conjecture of De Koninck regarding particular square values of the sum of divisors function, Journal of Number Theory, Vol. 137 (2014), pp. 50-66.
• Yong-Gao Chen and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Vol. 154 (2015), pp. 324-364.
• Jean-Marie De Koninck, Probelm 000:08, Western Number Theory Problems, 17 & 20 Dec 2000, edited by Gerry Myerson, p. 5.
• Jean-Marie De Koninck, Problem 10966, The American Mathematical Monthly, Vol. 109, No. 8 (2002), p. 759; Editorial comment, ibid., Vol. 111, No. 6 (2004), p. 536.
• Min Tang and Zhi-Jun Zhou, On a conjecture of De Koninck, INTEGERS, Vol. 18 (2018), Article #A60.
• Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
• Tomohiro Yamada, On a problem of De Koninck, Moscow Journal of Combinatorics and Number Theory, Vol. 10, No. 3 (2021), pp. 249-260; Correction, ibid., Vol. 10, No. 4 (2021), p. 339; arXiv preprint, arXiv:1906.10001 [math.NT], 2019-2021.

Crossrefs

Cf. A000203, A007947, A078615.

Subsequence of A175200.

Programs

Maple

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 :

Mathematica

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 *)

PARI

isok(n) = my(f=factor(n)); (sigma(f) % factorback(f[, 1])^2) == 0; \\ Michel Marcus, Nov 09 2020

Extensions

a(30)-a(35) from Donovan Johnson, Jan 14 2012