%I #47 Jan 05 2023 03:21:11

%S 1,4,12,27,28,124,279,508,1175,3125,7767,18125,32764,38205,53757,

%T 68013,86999,153575,171549,171875,271875,496341,524284,823543,1160541,

%U 1344573,1900557,1945233,2097148,2828375,4175037,4709853,5625261,6224013,7768031,10628469

%N Numbers k such that phi(k) - k = phi(k') - k', where k' is the arithmetic derivative of k and phi(k) is the Euler totient function.

%C A051674 is a subsequence of this sequence.

%H Amiram Eldar, <a href="/A239940/b239940.txt">Table of n, a(n) for n = 1..147</a>

%e The arithmetic derivative of 508 is 512, phi(508) = 252, phi(512) = 256 and 508 - 252 = 512 - 256 = 256, so 508 is a term.

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

%p for n from 1 to q do a:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);

%p if phi(a)-a=phi(n)-n then print(n); fi;

%p od; end: P(10^9);

%t d[1] = 0; d[n_] := n*Total[#2/#1 & @@@ FactorInteger[n]]; p[n_] := EulerPhi[n] - n; seqQ[n_] := p[d[n]] == p[n]; Select[Range[100000], seqQ] (* _Amiram Eldar_, Mar 31 2019 *)

%Y Cf. A000010, A003415, A051674, A239726.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Mar 31 2014

%E More terms from _Amiram Eldar_, Mar 31 2019