A070002 Numbers k such that phi(P(k)) - P(phi(k)) = 1, where P(k) is the largest prime factor of k.

45, 90, 135, 175, 180, 270, 350, 360, 405, 525, 540, 700, 720, 810, 875, 1050, 1080, 1215, 1400, 1440, 1573, 1575, 1620, 1750, 2100, 2160, 2430, 2625, 2800, 2880, 3146, 3150, 3240, 3500, 3645, 4200, 4320, 4375, 4719, 4725, 4860, 5250, 5491, 5600, 5760

Offset: 1

Author

Labos Elemer, May 07 2002

Keywords

• nonn

Comments

phi(P(k)) - P(phi(k)) = A000010(A006530(k)) - A006530(A000010(k)) = 1, where P(k) = largest prime factor of k. Value of commutator of phi and P functions at k equals 1.

Many but not all terms are divisible by 5.

Examples

			m = 77077 = 7*7*11*11*13*13 is here because P(m) = 13, phi(P(13)) = 12, phi(m) = 55440 = 2*2*2*2*3*3*5*7*11 with P(Phi(55440)) = 13 and the difference is 13 - 12 = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A006530, A068211, A070777, A070812, A070003, A070004, A007283, A070813, A070814, A070815, A070816.

Programs

Mathematica

pf[n_] := FactorInteger[n][[-1, 1]];
Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 1], Print[n]], {n, 3, 100000}]