A070814 - OEIS

%I #19 Jan 04 2017 10:26:43

%S 17,34,51,68,85,102,136,170,204,255,272,340,408,510,544,680,816,1020,

%T 1088,1360,1632,2040,2176,2720,3264,4080,4352,5440,6528,8160,8704,

%U 10880,13056,16320,17408,21760,26112,32640,34816,43520,52224,65280

%N Solutions to phi(gpf(x)) - gpf(phi(x)) = 14 = c are special multiples of 17, x = 17k, where greatest prime factors of factor k were observed from {2, 3, 5}, i.e., it is smaller than 17. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070815 for 254, A070816 for 65534. Gpf = greatest prime factor.

%C For n > 10, a(n) = 2a(n-4). First, it is easy to show that with i >= 0 and k,m in {0,1}, a(n) are of the form 2^i*3^k*5^m. Factoring this sequence reveals the regular pattern 2^i, 2^(i-2)*5, 2^(i-1)*3, 2^(i-3)*3*5, 2^(i+1), ... which obviously has the property a(n) = 2a(n-4) for n > 10. - Lambert Herrgesell (lambert.herrgesell(AT)googlemail.com), Jan 09 2007

%F For n > 10, a(n) = 2a(n-4) (conjectured). - _Ralf Stephan_, May 09 2004

%e For n = 32640 = 128*3*5*17, gpf(n) = 17, phi(n) = 16384, commutator[32640] = phi(17) - gpf(16384) = 16 - 2 = 14.

%t pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 14], Print[{n, n/17, pf[n/17]}]], {n, 3, 1000000}] (* Terms of sequence are n *)

%Y Cf. A000010, A000215, A006530, A007283, A070002, A070003, A070004, A070777, A070812, A070813.

%K nonn

%O 1,1

%A _Labos Elemer_, May 09 2002