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A255356 Composite squarefree numbers that are multiples of the root mean square of their prime factors. 0

%I

%S 1547,2737,4305,6545,13585,39997,52633,57505,65773,77441,93023,115855,

%T 202895,214415,285649,308865,315905,352495,352735,443555,449497,

%U 510229,510655,523439,611295,627095,650845,700321,722545,881705,936845,1088255,1103795,1392047

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square</a>.

%e 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.

%e 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.

%p with(numtheory); P:=proc(q) local a,b,c,k,n;

%p 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);

%p if type(n/b,integer) then print(n); fi; fi; od; end: P(10^9);

%t 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 *)

%Y Cf. A120944.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Feb 23 2015

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Last modified May 28 09:09 EDT 2022. Contains 354112 sequences. (Running on oeis4.)