cp's OEIS Frontend

A046762 Numbers k such that the sum of the squares of the divisors of k is divisible by k.

%I A046762 #36 Feb 17 2024 10:32:27
%S A046762 1,10,60,65,84,130,140,150,175,260,350,420,525,780,1050,1105,1820,
%T A046762 2100,2210,4420,4650,5425,5460,8840,10500,10850,13260,16275,19720,
%U A046762 20150,20737,21700,30225,30940,32045,32550,41474,45500,55250,57350,60450
%N A046762 Numbers k such that the sum of the squares of the divisors of k is divisible by k.
%C A046762 Compare with multiply perfect numbers A007691. Here Sum(divisors) is replaced by Sum(square of divisors).
%C A046762 Problem 11090 proves that this sequence is infinite. - _T. D. Noe_, Apr 18 2006
%C A046762 Cai, Chen, & Zhang prove that sigma_2(n)/n = b has only finitely many solutions for a given b, and hence (since this sequence is infinite) sigma_2(a(n))/a(n) is unbounded. - _Charles R Greathouse IV_, Jul 21 2016
%H A046762 Amiram Eldar, <a href="/A046762/b046762.txt">Table of n, a(n) for n = 1..3200</a> (terms 1..1000 from T. D. Noe)
%H A046762 Tianxin Cai, Deyi Chen, and Yong Zhang, <a href="http://www.worldscientific.com/doi/abs/10.1142/S1793042115500098">Perfect numbers and Fibonacci primes (I)</a>, Int. J. Number Theory 11, 159 (2015).
%H A046762 Florian Luca and John Ferdinands, <a href="http://www.jstor.org/stable/27641939">Problem 11090: Sometimes n divides sigma_k(n)</a>, Amer. Math. Monthly 113:4 (2006), pp. 372-373.
%e A046762 k = 65 = a(4), sigma(2,65) = 4420 = 65*68 = 68*k;
%e A046762 k = 1820 = a(17), the divisor-square sum is 4641000 = 2550*1820 = 2550*k.
%t A046762 Select[Range[70000],Divisible[DivisorSigma[2,#],#]&] (* _Harvey P. Dale_, Dec 15 2010 *)
%o A046762 (PARI) is(n)=sigma(n,2)%n==0 \\ _Charles R Greathouse IV_, Feb 04 2013
%Y A046762 Cf. A007691.
%K A046762 nonn
%O A046762 1,2
%A A046762 _Labos Elemer_