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Arithmetic numbers (A003601) that are not squares (A000290).
2

%I #13 Sep 08 2022 08:46:24

%S 3,5,6,7,11,13,14,15,17,19,20,21,22,23,27,29,30,31,33,35,37,38,39,41,

%T 42,43,44,45,46,47,51,53,54,55,56,57,59,60,61,62,65,66,67,68,69,70,71,

%U 73,77,78,79,83,85,86,87,89,91,92,93,94,95,96,97,99,101,102

%N Arithmetic numbers (A003601) that are not squares (A000290).

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

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

%C Corresponding values of A(m): 2, 3, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...

%C Corresponding values of G(m): sqrt(3), sqrt(5), sqrt(6), sqrt(7), sqrt(11), sqrt(13), sqrt(14), sqrt(15), sqrt(17), ...

%C Complement of A277793 with respect to A003601.

%t Select[Range[100], !IntegerQ @ Sqrt[#] && Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] (* _Amiram Eldar_, Oct 20 2019 *)

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

%o (PARI) isA328557(n) = (!issquare(n)&&!(sigma(n)%numdiv(n))); \\ _Antti Karttunen_, Oct 19 2019

%Y Cf. A000005, A000203, A000290, A003601, A277793, A328558, A328559.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Oct 19 2019