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Numbers m that are neither arithmetic (A003601) nor squares (A000290).
2

%I #7 Sep 08 2022 08:46:24

%S 2,8,10,12,18,24,26,28,32,34,40,48,50,52,58,63,72,74,75,76,80,82,84,

%T 88,90,98,104,106,108,112,117,120,122,124,128,130,136,146,148,152,156,

%U 160,162,170,171,172,175,176,178,180,192,194,200,202,208,216,218,226

%N Numbers m that are neither arithmetic (A003601) nor squares (A000290).

%C Numbers m such that neither the arithmetic mean of the divisors of m nor the geometric mean of the divisors of m is an integer.

%C Numbers m such that neither A(m) = A000203(m) / A000005(m) nor G(m) = sqrt(m) is an integer.

%C Corresponding values of A(m): 3/2, 15/4, 9/2, 14/3, 13/2, 15/2, 21/2, 28/3, 21/2, 27/2, 45/4, 62/5, 31/2, 49/3, 45/2, ...

%C Corresponding values of G(m): sqrt(2), sqrt(8), sqrt(10), sqrt(12), sqrt(18), sqrt(24), sqrt(26), sqrt(28), sqrt(32), ...

%t Select[Range[230], !IntegerQ@Sqrt[#] && !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] (* _Amiram Eldar_, Nov 01 2019 *)

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

%Y Cf. A000005, A000203, A000290, A003601, A328557, A328558.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Oct 31 2019