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

1,1

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, ...

Links

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

Example

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

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]

Prog

(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

Crossrefs

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

Sequence in context: A028331 A279729 A357946 * A113806 A105775 A159469

Adjacent sequences: A309650 A309651 A309652 * A309654 A309655 A309656

Keyword

nonn

Author

Amiram Eldar, Aug 11 2019

Status

approved