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

1,2

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).

Links

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

Example

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).

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 *)

Prog

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

Crossrefs

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

Sequence in context: A173409 A057009 A140480 * A002315 A141813 A088165

Adjacent sequences: A327052 A327053 A327054 * A327056 A327057 A327058

Keyword

nonn

Author

Jaroslav Krizek, Oct 07 2019

Status

approved