A244439 - OEIS

%I #25 Jun 08 2020 17:09:11

%S 5,55,56,123,135,147,175,304,351,644,1015,2464,19304,61131,162524,

%T 476671,567644,712724,801944,2435488,3346399,3885056,4555999,8085560,

%U 8369360,12516692,22702119,29628800,83884031,83994624,84789247,354812535,860616295,1091535704

%N Numbers n such that phi(n)*sigma(n) = phi(n+1)*sigma(n+1).

%C Since both numbers 55 and 56 are in the sequence we have sigma(55)*phi(55) = sigma(56)*phi(56) = sigma(57)*phi(57). It seems that 56 is the only number n which has the nice property sigma(n-1)*phi(n-1) = sigma(n)*phi(n) = sigma(n+1)*phi(n+1).

%C Up to n < 6*10^11 the similar equation phi(n)*sigma(n+1) = phi(n+1)*sigma(n) is satisfied only by n = 696003. - _Giovanni Resta_, Jun 08 2020

%H Giovanni Resta, <a href="/A244439/b244439.txt">Table of n, a(n) for n = 1..52</a> (terms < 10^13, first 40 terms from Jens Kruse Andersen)

%e 5 is in the sequence because sigma(5)*phi(5) = sigma(6)*phi(6) = 24.

%e 55 is in the sequence because sigma(55)*phi(55) = sigma(56)*phi(56) = 2880.

%p with(numtheory): A244439:=n->`if`(phi(n)*sigma(n) = phi(n+1)*sigma(n+1), n, NULL): seq(A244439(n), n=1..10^4); # _Wesley Ivan Hurt_, Aug 16 2014

%t Select[Range[10^5], Equal @@ (EulerPhi[{#, # + 1}] DivisorSigma[1, {#, # + 1}]) &] (* _Giovanni Resta_, Jun 08 2020 *)

%o (PARI)

%o for(n=1,10^6,s=eulerphi(n)*sigma(n);if(s==eulerphi(n+1)*sigma(n+1),print1(n,", "))) \\ _Derek Orr_, Aug 14 2014

%Y Cf. A000010, A000203, A145749.

%K nonn

%O 1,1

%A _Farideh Firoozbakht_, Aug 14 2014

%E More terms from _Jens Kruse Andersen_, Aug 16 2014