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

1,2

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

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

Example

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.

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 *)

Crossrefs

Cf. A000290, A003601, A107924, A107925.

Sequence in context: A256074 A369565 A016922 * A147608 A258060 A039686

Adjacent sequences: A277790 A277791 A277792 * A277794 A277795 A277796

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Oct 31 2016

Status

approved