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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 11 2019
STATUS
approved