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.

1, 3, 2, 14, 190, 6, 78, 42, 30, 570, 16770, 210, 1102290, 2730, 67830, 43890, 133707210, 746130, 27606810, 16546530, 9699690, 417086670, 3828438543930, 8720021310, 705196562070

Offset: 1

Author

Altug Alkan, Aug 17 2017

Keywords

• nonn

Comments

Also a(n) is the smallest squarefree number k such that sigma(k) = n*phi(k), or 0 if no such k exists.

It is conjectured that A055234(n) > 0 for each n. Is a(n) > 0 for all values of n?

10^12 < a(26) <= 50353622409090. - Giovanni Resta, Aug 18 2017

Examples

			a(4) = 14 since psi(14) / phi(14) = 24 / 6 = 4 and 14 is the least number with this property.
		

Crossrefs

Cf. A000010, A001615, A055234.

Programs

Mathematica

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 *)

PARI

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
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

Extensions

a(23)-a(25) from Giovanni Resta, Aug 18 2017