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%I #22 Aug 17 2023 22:35:37
%S 0,-1,2,0,3,-1,-1,2,5,0,9,3,2,-1,14,-1,15,2,3,5,11,0,-1,9,-1,3,21,2,
%T 25,-1,5,14,3,-1,33,15,9,2,35,3,35,5,1,11,23,0,-1,-1,14,9,39,-1,5,3,
%U 15,21,29,2,55,25,3,-1,9,5,55,14,11,3,63,-1,69,33,-1,15,5,9,65,2,-1,35,41,3,14,35,21,5,77,1,9,11,25,23,15,0,93,-1,5
%N a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).
%C Value of commutator[A000010, A006530] at n.
%H Michael De Vlieger, <a href="/A070812/b070812.txt">Table of n, a(n) for n = 3..10000</a>
%H Michael De Vlieger, <a href="/A070812/a070812.png">Log log scatterplot of a(n)+2</a>, n = 3..10^5.
%F a(n) = A070777(n) - A068211(n).
%e Cases of n when a(n) = 1, -1, 2 or 0 are listed in A070002, A070003, A070004, A007283 respectively. Further regular solutions: if a(n)=3, then n=7k, where k has prime divisors < 7; if a(n)=5, then n=11k, where k has no prime divisors >=11; if a(n)=25, then mostly (not always!) n=31k ...
%t pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Table[EulerPhi[pf[u]]-pf[EulerPhi[u]], {u, 3, 128}]
%o (PARI) gpf(n)=my(f=factor(n)[,1]);f[#f]
%o a(n)=gpf(n)-gpf(eulerphi(n))-1 \\ _Charles R Greathouse IV_, Feb 19 2013
%Y Cf. A000010, A006530, A070777, A068211, A070002, A070003, A070004, A007283.
%K sign
%O 3,3
%A _Labos Elemer_, May 09 2002