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A272857 Least k>1 such that the Euler totient function of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

%I A272857 #32 May 27 2024 07:14:40
%S A272857 3,3,13,61,61,421,2521,2521,2521,55441,55441,4324321,4324321,4324321,
%T A272857 4324321,85765681,85765681,232792561,232792561,232792561,232792561
%N A272857 Least k>1 such that the Euler totient function of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).
%C A272857 a(22) <= 10708457761. - _Amiram Eldar_, May 27 2024
%e A272857 phi(3) / d(3) = 2 / 2 = 1, phi(3^2) / d(3^2) = 6 / 3 = 2 but phi(3^3) / d(3^3) = 18 / 4 = 9 / 2;
%e A272857 phi(13) / d(13) = 12 / 2 = 6, phi(13^2) / d(13^2) = 156 / 3 = 52, phi(13^3) / d(13^3) = 2028 / 4 = 507 but phi(13^4) / d(13^4) = 26364 / 5.
%p A272857 with(numtheory): P:= proc(q) local a, j, k, ok, p; global n; a:=2;
%p A272857 for k from 1 to q do for n from a to q do ok:=1;
%p A272857 for j from 1 to k do if not type(phi(n^j)/tau(n^j), integer) then ok:=0; break; fi; od;
%p A272857 if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);
%Y A272857 Cf. A000005, A000010, A070858, A272980, A272981.
%K A272857 nonn,more
%O A272857 1,1
%A A272857 _Paolo P. Lava_, May 12 2016