A070004 Numbers of the form 5*2^n or 5*3*2^n; a(n) = 5*A029744(n).

5, 10, 15, 20, 30, 40, 60, 80, 120, 160, 240, 320, 480, 640, 960, 1280, 1920, 2560, 3840, 5120, 7680, 10240, 15360, 20480, 30720, 40960, 61440, 81920, 122880, 163840, 245760, 327680, 491520, 655360, 983040, 1310720, 1966080, 2621440

Offset: 1

Author

Labos Elemer, May 07 2002

Keywords

• nonn
• easy

Comments

Old name was: Numbers n such that phi(P(n)) - P(phi(n)) = 2, where P(x)=largest prime factor of x, or A000010(A006530(n))-A006530(A000010(n))=2.

Solutions to phi(P(x))-P(phi(x))=c, presence or absence of special prime factors in x are usually derivable.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,2).

Crossrefs

Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070003, A007283, A070813-A070816.

Programs

Mathematica

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2]; Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 2], Print[n]], {n, 3, 1000000}]
Union[Flatten[Table[2^n {5,15},{n,0,20}]]] (* or *) Join[ {5}, LinearRecurrence[ {0,2},{10,15},40]] (* Harvey P. Dale, Dec 23 2014 *)

PARI

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
is(n)=eulerphi(gpf(n))-gpf(eulerphi(n))==2 \\ Charles R Greathouse IV, Feb 19 2013

Formula

a(n) = 5*A029744(n); numbers of the forms 5*2^n and 15*2^n.

G.f.: 5*x*(x+1)^2/(1-2*x^2). - Ralf Stephan, Jul 15 2013

Sum_{n>=1} 1/a(n) = 8/15. - Amiram Eldar, Jan 02 2021

Extensions

Simpler name by Joerg Arndt, Jul 16 2013