A279814 Numbers n such that the average of the squares of the proper divisors of n is an integer.

2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 101, 103, 106, 107, 109, 113, 115, 118, 119, 121, 122, 125, 127, 131, 133, 134, 137, 139, 140, 142, 143, 145, 146, 149

Offset: 1

Author

Ilya Gutkovskiy, Dec 19 2016

Keywords

• nonn
• easy

Comments

Numbers n such that number of proper divisors of n (A032741) divides sum of squares of proper divisors of n (A067558).

All the prime numbers are in this sequence.

Examples

			8 is in the sequence because 8 has 3 proper divisors {1,2,4}, 1^2 + 2^2 + 4^2 = 21 and 3 divides 21.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A020486, A023884, A032741, A067558.

Programs

Mathematica

Select[Range[150], Mod[DivisorSigma[2, #1] - #1^2, DivisorSigma[0, #1] - 1] == 0 &]
Select[Range[200],IntegerQ[Mean[Most[Divisors[#]]^2]]&] (* Harvey P. Dale, Jul 26 2019 *)

PARI

is(n)=my(d=divisors(n)); d=apply(k->k^2, d[1..#d-1]); n>1 && sum(i=1,#d,d[i])%#d==0 \\ Charles R Greathouse IV, Dec 19 2016