A309653 Composite numbers k such that phi(k) * psi(k) + 1 is a perfect square, where phi is the Euler totient function (A000010) and psi is the Dedekind psi function (A001615).

6, 8, 20, 22, 33, 69, 82, 156, 171, 190, 198, 295, 354, 451, 581, 664, 1119, 1142, 1175, 1184, 2812, 2893, 4043, 4163, 4262, 4581, 5090, 6964, 7018, 12977, 14927, 15026, 15753, 19105, 22828, 22926, 25132, 25369, 28919, 29746, 38013, 39146, 47932, 74666, 80375

Offset: 1

Author

Amiram Eldar, Aug 11 2019

Keywords

• nonn

Comments

For all primes p, phi(p) * psi(p) + 1 = (p-1) * (p+1) + 1 = p^2 is a perfect square.

The squarefree terms of this sequence are common with the squarefree terms of A015709 since sigma(k) = psi(k) for squarefree numbers k.

If p is in A096147 then 2*p is in this sequence.

If p is in A078699 (prime p such that p^2 - 1 is a triangular number) then 3*p is in this sequence.

If p is a prime such that 2*p^2 - 2*p - 1 is also a prime then p*(2*p^2 - 2*p - 1) is in this sequence. These primes are 2, 3, 7, 13, 19, 37, 79, 103, 127, 199, 241, 307, 313, 331, 337, ...

Examples

			8 is in the sequence since phi(8) * psi(8) + 1 = 4 * 12 + 1 = 49 = 7^2 is a perfect square.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..200 from Robert Israel)

Crossrefs

Cf. A000010, A001615, A015709, A078699, A096147.

Programs

Maple

filter:= proc(n)
local t;
if isprime(n) then return false fi;
issqr(1 + mul(t[1]^(2*t[2]-2)*(t[1]^2-1),t=ifactors(n)[2]))
end proc:
select(filter, [$2..10^5]); # Robert Israel, Aug 13 2019

Mathematica

f[p_, e_] := (p^e - p^(e - 1))*(p^e + p^(e - 1)); psiphi[n_] := Times @@ (f @@@ FactorInteger[n]); aQ[n_] := CompositeQ[n] && IntegerQ@Sqrt[psiphi[n] + 1]; Select[Range[1000], aQ]

PARI

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = !isprime(k) && issquare(eulerphi(k)*mypsi(k) + 1); \\ Michel Marcus, Aug 11 2019