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A327055 Numbers m such that the arithmetic mean and the quadratic mean (the root mean square) of the divisors of m are both integers. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 25 12:12 EDT 2022. Contains 356984 sequences. (Running on oeis4.)