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

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

Cf. A007691.

Sequence in context: A213346 A140890 A055714 * A066290 A065641 A121874

Adjacent sequences: A046759 A046760 A046761 * A046763 A046764 A046765

Keyword

nonn

Author

Labos Elemer

Status

approved