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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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: A199884 A028331 A279729 * A113806 A105775 A159469

Adjacent sequences:  A309650 A309651 A309652 * A309654 A309655 A309656

KEYWORD

nonn

AUTHOR

Amiram Eldar, Aug 11 2019

STATUS

approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)