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A082344 Denominator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).

%I A082344 #14 Dec 04 2021 12:33:33
%S A082344 1,1,1,1,1,6,1,4,3,10,1,12,1,14,15,2,1,9,1,20,21,22,1,8,5,26,3,28,1,3,
%T A082344 1,16,33,34,35,18,1,38,39,40,1,7,1,44,45,46,1,48,7,25,51,52,1,54,55,
%U A082344 56,57,58,1,5,1,62,63,16,65,33,1,68,69,5,1,6,1,74,75,76,77,13,1,80,27,82,1,6
%N A082344 Denominator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).
%C A082344 Numerator is A082343(n) = A001414(n)/A082299(n).
%H A082344 Antti Karttunen, <a href="/A082344/b082344.txt">Table of n, a(n) for n = 1..65537</a>
%F A082344 a(n) = n/A082299(n).
%e A082344 n=200: (2+2+2+5+5)/(2*2*2*5*5) = 16/(2*2*2*5*5) = (2*2*2*2)/(2*2*2*5*5) = 2/25, therefore a(200)=25, A082343(200)=2.
%t A082344 sopd[n_]:=Module[{f=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]}, Denominator[ Total[f]/n]]; Array[sopd,90] (* _Harvey P. Dale_, Jul 24 2018 *)
%t A082344 sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
%t A082344 a[n_] := Denominator[sopfr[n]/n];
%t A082344 Array[a, 100] (* _Jean-François Alcover_, Dec 03 2021 *)
%o A082344 (PARI)
%o A082344 A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
%o A082344 A082299(n) = gcd(n, A001414(n));
%o A082344 A082344(n) = (n/A082299(n)); \\ _Antti Karttunen_, Mar 04 2018
%Y A082344 Cf. A001414, A082299, A082343.
%K A082344 nonn,frac
%O A082344 1,6
%A A082344 _Reinhard Zumkeller_, Apr 09 2003