A327055 Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers.

1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, 34697, 46655, 66815, 68593, 68985, 125255, 155287, 182665, 242879, 273265, 380511, 391345, 404055, 421655, 627215, 730145, 814463, 823537, 876785, 1069895, 1087009, 1166399, 1204281, 1256489

Offset: 1

Author

Jaroslav Krizek, Oct 07 2019

Keywords

• nonn

Comments

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

Intersection of A003601 and A140480.

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).

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

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).

Examples

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..7391 (terms < 10^13)

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)