OFFSET
1,1
LINKS
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
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Feb 23 2015
STATUS
approved