A328558 Squares (A000290) that are not arithmetic numbers (A003601).

4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 196, 225, 256, 289, 324, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 1024, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1681, 1764, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136

Offset: 1

Author

Jaroslav Krizek, Oct 19 2019

Keywords

• nonn

Comments

Numbers m such that the geometric mean of the divisors of m is an integer but the arithmetic mean of the divisors of m is not an integer.

Numbers m such that G(m) = sqrt(m) is an integer but A(m) = A000203(m) / A000005(m) is not an integer.

Corresponding values of G(m): 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, ...

Corresponding values of A(m): 7/3, 13/3, 31/5, 31/3, 91/9, 127/7, 121/5, 217/9, 133/3, 403/15, 133/3, 403/9, ...

Complement of A277793 with respect to A000290.

Crossrefs

Cf. A000005, A000203, A000290, A003601, A277793, A328557, A328559.

Programs

Magma

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(Sqrt(m))]

Mathematica

Select[Range[56]^2, !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] (* Amiram Eldar, Oct 20 2019 *)

PARI

isA328558(n) = (issquare(n)&&(sigma(n)%numdiv(n))); \\ Antti Karttunen, Oct 19 2019