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Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers.
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%I #12 Sep 08 2022 08:46:24

%S 1,7,41,239,287,1673,3055,6665,9545,9799,9855,21385,26095,34697,46655,

%T 66815,68593,68985,125255,155287,182665,242879,273265,380511,391345,

%U 404055,421655,627215,730145,814463,823537,876785,1069895,1087009,1166399,1204281,1256489

%N Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers.

%C Numbers m such that A000203(m) / A000005(m) and sqrt(A001157(m) / A000005(m)) are both integers.

%C Intersection of A003601 and A140480.

%C Sequence deviates from A140480 (RMS numbers); first deviation is at a(461), a(461) = 2226133343. Number A140480(461) = 2217231104 is the first RMS number that are not arithmetic (see A327056 for such numbers).

%C Corresponding values of A000203(a(n)) / A000005(a(n)): 1, 4, 21, 120, 84, 480, 504, 1056, 1512, 2520, 1110, 2016, 4158, ...

%C Corresponding values of sqrt(A001157(a(n)) / A000005(a(n))): 1, 5, 29, 169, 145, 845, 1105, 2405, 3445, 4901, 2665, 5525, ... (sequence deviates from A141812).

%H Giovanni Resta, <a href="/A327055/b327055.txt">Table of n, a(n) for n = 1..7391</a> (terms < 10^13)

%e Number 41 is a term because sigma(41) / tau(41) = 42 / 2 = 21 and sqrt((1^2 + 41^2) / tau(41) ) = sqrt(1682 / 2) = 29.

%e Values of means of the first RMS number 2217231104 that is not in the sequence: 418652080/9 (noninteger) and 247511537 (integer).

%t aQ[n_] := IntegerQ[DivisorSigma[1, n]/(d = DivisorSigma[0, n])] && IntegerQ @ Sqrt[DivisorSigma[2, n]/d]; Select[Range[10^5], aQ] (* _Amiram Eldar_, Oct 07 2019 *)

%o (Magma) [m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(Sqrt(&+[d^2: d in Divisors(m)] / NumberOfDivisors(m)))]

%Y Cf. A000005, A000203, A001157, A140480, A141812, A158298, A158299, A327056.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Oct 07 2019