A273593 - OEIS

%I #12 Jun 08 2016 02:07:24

%S 310,364,5497,11592,40752,494328,3030109,3554874,128984839

%N Numbers which are equal to the sum of the aliquot parts of the aliquot parts of their Euler totient function.

%C Fixed points of the transform n -> Sum_{i=1..k}{sigma(d_i)-d_i}, where d_i are the aliquot parts of phi(n). The first two couples connected by this transform are (109, 186) and (8826, 9198).

%C a(10) > 2.5*10^9. - _Giovanni Resta_, May 30 2016

%e Euler totient function of 310 is 120 and its aliquot parts are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40 and 60. Then: sigma(1) - 1 = 0, sigma(2) - 2 = 1, sigma(3) - 3 = 1, sigma(4) - 4 = 3, sigma(5) - 5 = 1, sigma(6) - 6 = 6, sigma(8) - 8 = 7, sigma(10) - 10 = 8, sigma(12) - 12 = 16, sigma(15) - 15 = 9, sigma(20) - 20 = 22, sigma(24) - 24 = 36, sigma(30) - 30 = 42, sigma(40) - 40 = 50, sigma(60) - 60 = 108 and 0 + 1 + 1 + 3 + 1 + 6 + 7 + 8 + 16 + 9 + 22 + 36 + 42 + 50 + 108 = 310.

%p with(numtheory): P:=proc(q) local a,k,n;

%p for n from 1 to q do a:=sort([op(divisors(phi(n)))]);

%p if n=add(sigma(a[k])-a[k],k=1..nops(a)-1) then print(n); fi; od; end: P(10^9);

%o (PARI) is(n)=my(t=eulerphi(n)); sumdiv(t,d,if(d<t, sigma(d)-d))==n \\ _Charles R Greathouse IV_, Jun 08 2016

%Y Cf. A000010, A000203, A001065.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, May 26 2016

%E a(7)-a(9) from _Giovanni Resta_, May 30 2016