A343011 Integers whose variance of their divisors is also an integer.

1, 3, 5, 7, 11, 13, 15, 17, 19, 20, 21, 23, 24, 27, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 49, 51, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 71, 73, 77, 79, 83, 85, 87, 88, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 116, 119, 120, 123

Offset

1,2

Comments

The sequence contains all odd prime numbers p, since they have N=2 divisors whose mean xbar is an integer and the sum of their squared mean deviations SS is even; therefore, their variance Var(p)=SS/N is an integer.

k is a term if and only if A000005(k)^2 divides (A000005(k)*A001157(k)-A000203(k)^2). - Chai Wah Wu, Apr 11 2021

Links

Table of n, a(n) for n=1..62.

Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.

Example

a(10)=20 has six divisors d={1,2,4,5,10,20} whose variance is Var(d)=42, an integer.

Mathematica

Select[Range[500], IntegerQ[Mean[(Divisors[#] - Mean[Divisors[#]])^2]]&]

Prog

(PARI) isok(m) = my(d=divisors(m)); denominator(norml2(d)/#d - (vecsum(d)/#d)^2) == 1; \\ Michel Marcus, Apr 03 2021
(Python)
from sympy import divisor_sigma
A343011_list = [n for n in range(1, 10**3) if (divisor_sigma(n, 0)*divisor_sigma(n, 2)-divisor_sigma(n, 1)**2) % divisor_sigma(n, 0)**2 == 0] # Chai Wah Wu, Apr 11 2021

Crossrefs

Cf. A000005, A000203, A001157, A003601, A065091.

Sequence in context: A371168 A370811 A070087 * A100933 A325128 A352830

Adjacent sequences: A343008 A343009 A343010 * A343012 A343013 A343014

Keyword

nonn

Author

Alexander M. Kerr, Apr 02 2021

Status

approved