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A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

%I A066086 #26 Dec 05 2018 08:00:22
%S A066086 1,1,2,1,2,2,2,1,2,2,2,2,2,6,8,1,2,2,2,2,4,2,2,2,2,6,2,6,2,8,2,1,4,2,
%T A066086 24,2,2,6,8,2,2,12,2,2,8,2,2,2,2,2,8,6,2,2,8,6,4,2,2,8,2,6,4,1,12,4,2,
%U A066086 2,4,24,2,2,2,6,8,6,12,24,2,2,2,2,2,12,4,6,8,2,2,8,8,2,4,2,24,2,2,6,4
%N A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).
%C A066086 Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).
%H A066086 Antti Karttunen, <a href="/A066086/b066086.txt">Table of n, a(n) for n = 1..23374</a>
%H A066086 Antti Karttunen, <a href="/A066086/a066086.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%F A066086 a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).
%F A066086 a(n) = A322360(n) / A322359(n). - _Antti Karttunen_, Dec 04 2018
%t A066086 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
%t A066086 a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* _Amiram Eldar_, Dec 05 2018 *)
%o A066086 (PARI) a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ _Charles R Greathouse IV_, Feb 14 2013
%Y A066086 Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.
%K A066086 nonn
%O A066086 1,3
%A A066086 _Labos Elemer_, Dec 04 2001
%E A066086 Name edited, part of the old name transferred to the formula section by _Antti Karttunen_, Dec 04 2018