A255356 Composite squarefree numbers that are multiples of the root mean square of their prime factors.

1547, 2737, 4305, 6545, 13585, 39997, 52633, 57505, 65773, 77441, 93023, 115855, 202895, 214415, 285649, 308865, 315905, 352495, 352735, 443555, 449497, 510229, 510655, 523439, 611295, 627095, 650845, 700321, 722545, 881705, 936845, 1088255, 1103795, 1392047

Offset

1,1

Links

Table of n, a(n) for n=1..34.

Eric Weisstein's World of Mathematics, Root-Mean-Square.

Example

Prime factors of 1547 are 7, 13, 17. Their root mean square is sqrt((7^2 + 13^2 + 17^2) / 3) = sqrt((49 + 169 + 289) / 3) = sqrt(507 / 3) = sqrt(169) = 13 and 1547 / 13 = 119.
Prime factors of 2737 are 7, 17, 23. Their root mean square is sqrt((7^2 + 17^2 + 23^2) / 3) = sqrt((49 + 289 + 529) / 3) = sqrt(867 / 3) = sqrt(289) = 17 and 2737 / 17 = 161.

Maple

with(numtheory); P:=proc(q) local a, b, c, k, n;
for n from 2 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2]; c:=add(a[k][2], k=1..nops(a)); b:=sqrt(add(a[k][2]*a[k][1]^2, k=1..nops(a))/c);
if type(n/b, integer) then print(n); fi; fi; od; end: P(10^9);

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, Length[f] > 1 && AllTrue[f[[;; , 2]], # == 1 &] && Divisible[n, RootMeanSquare[f[[;; , 1]]]]]; Select[Range[10^5], q] (* Amiram Eldar, Feb 24 2021 *)

Crossrefs

Cf. A120944.

Sequence in context: A090209 A283154 A157347 * A336221 A020408 A259424

Adjacent sequences: A255353 A255354 A255355 * A255357 A255358 A255359

Keyword

nonn

Author

Paolo P. Lava, Feb 23 2015

Status

approved