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Numbers k such that 1/phi(x) + 1/phi(y) = 1/phi(k), for some x + y = k and phi(k) is the Euler totient function of k.
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%I #32 Jul 22 2019 18:42:22

%S 1890,2100,2310,3780,5250,7770,10080,11310,11550,11880,12180,13230,

%T 13650,13860,14190,14910,15750,17640,18060,19950,20460,20790,21630,

%U 22050,22110,23100,24090,24180,24570,25410,25620,25830,26070,27090,27510,27720,28980,29040,29400

%N Numbers k such that 1/phi(x) + 1/phi(y) = 1/phi(k), for some x + y = k and phi(k) is the Euler totient function of k.

%C All terms appear to be multiples of 30.

%C Terms that are not divisible by 30: 70224, 72072, 96558, 114114, 122892, 156156, 166782, 184338, 191268, ... - _Amiram Eldar_, Jul 22 2019

%H Amiram Eldar, <a href="/A279621/b279621.txt">Table of n, a(n) for n = 1..1000</a>

%e 1890 = 817 + 1073 and 1/phi(817) + 1/phi(1073) = 1/756 + 1/1008 = 1/432 = 1/phi(1890).

%e The first term with more than one solution is 14190:

%e 14190 = 6319 + 7871 and 1/phi(6319) + 1/phi(7871) = 1/6160 + 1/7392 = 1/3360 = 1/phi(14190).

%e 14190 = 6443 + 7747 and 1/phi(6443) + 1/phi(7747) = 1/6048 + 1/7560 = 1/3360 = 1/phi(14190).

%p with(numtheory): P:= proc(q) local k,n; for n from 1 to q do

%p for k from 1 to trunc(n/2) do if 1/phi(k)+1/phi(n-k)=1/phi(n)

%p then print(n); break; fi; od; od; end: P(10^6);

%t aQ[n_] := Module[{k = 1, r = 1/EulerPhi[n]}, While[2*k <= n && 1/EulerPhi[k] + 1/EulerPhi[n - k] != r, k++]; 2*k <= n]; (* _Amiram Eldar_, Jul 22 2019 *)

%Y Cf. A000010.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Dec 19 2016