A046762 - OEIS

A046762

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

1, 10, 60, 65, 84, 130, 140, 150, 175, 260, 350, 420, 525, 780, 1050, 1105, 1820, 2100, 2210, 4420, 4650, 5425, 5460, 8840, 10500, 10850, 13260, 16275, 19720, 20150, 20737, 21700, 30225, 30940, 32045, 32550, 41474, 45500, 55250, 57350, 60450

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OFFSET

1,2

COMMENTS

Compare with multiply perfect numbers A007691. Here Sum(divisors) is replaced by Sum(square of divisors).

Problem 11090 proves that this sequence is infinite. - T. D. Noe, Apr 18 2006

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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..1000 from T. D. Noe)

Tianxin Cai, Deyi Chen, and Yong Zhang, Perfect numbers and Fibonacci primes (I), Int. J. Number Theory 11, 159 (2015).

Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.

EXAMPLE

k = 65 = a(4), sigma(2,65) = 4420 = 65*68 = 68*k;

k = 1820 = a(17), the divisor-square sum is 4641000 = 2550*1820 = 2550*k.

MATHEMATICA

Select[Range[70000], Divisible[DivisorSigma[2, #], #]&] (* Harvey P. Dale, Dec 15 2010 *)

PROG

(PARI) is(n)=sigma(n, 2)%n==0 \\ Charles R Greathouse IV, Feb 04 2013

CROSSREFS