A070812 a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).

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, 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, 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

Offset: 3

Author

Labos Elemer, May 09 2002

Keywords

• sign

Comments

Value of commutator[A000010, A006530] at n.

Examples

			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 ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..10000
• Michael De Vlieger, Log log scatterplot of a(n)+2, n = 3..10^5.

Crossrefs

Cf. A000010, A006530, A070777, A068211, A070002, A070003, A070004, A007283.

Programs

Mathematica

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Table[EulerPhi[pf[u]]-pf[EulerPhi[u]], {u, 3, 128}]

PARI

gpf(n)=my(f=factor(n)[,1]);f[#f]
a(n)=gpf(n)-gpf(eulerphi(n))-1 \\ Charles R Greathouse IV, Feb 19 2013

Formula

a(n) = A070777(n) - A068211(n).