A277793 Numbers k such that the arithmetic and geometric means of the divisors of k are both integers.

1, 49, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 8281, 9409, 10609, 11881, 14641, 16129, 17689, 19321, 22801, 24649, 26569, 32761, 37249, 39601, 44521, 47089, 49729, 52441, 58081, 61009, 67081, 73441, 76729, 80089, 87616, 90601, 94249, 97969, 109561, 113569, 121801, 134689

Offset: 1

Author

Ilya Gutkovskiy, Oct 31 2016

Keywords

• nonn
• easy

Comments

Intersection of A000290 and A003601.

Union of squares of A107924 and squares of A107925.

The squares of the primes == 1 (mod 6), squares of A002476, are a subsequence: 49, 169, 361,... - R. J. Mathar, May 19 2020

Examples

			a(2) = 49 because 49 has 3 divisors {1,7,49} therefore (1 + 7 + 49)/3 = 19 and (1*7*49)^(1/3) = 7 are both integers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor
• Wikipedia, Arithmetic number
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000290, A003601, A107924, A107925.

Programs

Mathematica

Select[Range[140000], Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] && Mod[DivisorSigma[0, #1], 2] == 1 & ]
Select[Range[150000],AllTrue[{Mean[Divisors[#]],GeometricMean[ Divisors[ #]]}, IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 21 2018 *)