cp's OEIS Frontend

A291051 a(n) is the smallest number k such that psi(k) = n*phi(k) where psi(k) is Dedekind psi function (A001615) and phi(k) is Euler totient function (A000010), or 0 if no such k exists.

%I A291051 #28 Sep 19 2017 12:02:22
%S A291051 1,3,2,14,190,6,78,42,30,570,16770,210,1102290,2730,67830,43890,
%T A291051 133707210,746130,27606810,16546530,9699690,417086670,3828438543930,
%U A291051 8720021310,705196562070
%N A291051 a(n) is the smallest number k such that psi(k) = n*phi(k) where psi(k) is Dedekind psi function (A001615) and phi(k) is Euler totient function (A000010), or 0 if no such k exists.
%C A291051 Also a(n) is the smallest squarefree number k such that sigma(k) = n*phi(k), or 0 if no such k exists.
%C A291051 It is conjectured that A055234(n) > 0 for each n. Is a(n) > 0 for all values of n?
%C A291051 10^12 < a(26) <= 50353622409090. - _Giovanni Resta_, Aug 18 2017
%e A291051 a(4) = 14 since psi(14) / phi(14) = 24 / 6 = 4 and 14 is the least number with this property.
%t A291051 psi[n_] := n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; f[n_] := Block[{k = 1}, While[ n*EulerPhi[k] != psi[k], k++]; k]; Array[f, 22] (* _Robert G. Wilson v_, Sep 15 2017 *)
%o A291051 (PARI) a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
%o A291051 a(n) = {my(k = 1); while (n*eulerphi(k) != a001615(k), k++); k; } \\ _Altug Alkan_, Aug 17 2017, after _Charles R Greathouse IV_ at A001615
%Y A291051 Cf. A000010, A001615, A055234.
%K A291051 nonn
%O A291051 1,2
%A A291051 _Altug Alkan_, Aug 17 2017
%E A291051 a(23)-a(25) from _Giovanni Resta_, Aug 18 2017