A247136 Numbers for which the root mean square of nontrivial divisors is an integer.

4, 9, 25, 49, 119, 121, 161, 169, 289, 343, 361, 369, 527, 529, 711, 721, 833, 841, 959, 961, 1081, 1127, 1241, 1369, 1681, 1695, 1767, 1849, 2047, 2209, 2809, 3281, 3335, 3481, 3553, 3713, 3721, 4207, 4489, 4633, 4681, 5041, 5047, 5215, 5329, 6241, 6713, 6887

Offset

1,1

Comments

All the squares of prime numbers (A001248) have this property but there are other numbers (A247137): 119,161,343,369,527,711,721,833,959,1081...

Links

Daniel Lignon and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 316 terms from Lignon)

Formula

Trivially a(n) << n^2 log^2 n. - Charles R Greathouse IV, Nov 20 2014

Example

119 is a term because the nontrivial divisors of 119 are [7,17] and sqrt((7^2+17^2)/2)= 13 : it's an integer.

Mathematica

Select[Range[2, 100000], (Not[PrimeQ[#]] && IntegerQ[ RootMeanSquare[ Rest[ Most[ Divisors[#]]]]])&]

Prog

(PARI) integralRMS(v)=my(t=norml2(v)/#v); denominator(t)==1 && issquare(t)
is(n)=my(d=divisors(n)); #d>2 && integralRMS(d[2..#d-1]) \\ Charles R Greathouse IV, Nov 20 2014

Crossrefs

Cf. A140480 (numbers for which the root mean square of all divisors is an integer), A247136 (numbers for which the root mean square of proper divisors is an integer) and A023886 (numbers for which the arithmetic mean of nontrivial divisors is an integer).

Sequence in context: A344701 A182988 A158144 * A158145 A082180 A246131

Adjacent sequences: A247133 A247134 A247135 * A247137 A247138 A247139

Keyword

nonn

Author

Daniel Lignon, Nov 20 2014

Status

approved