A015709 -id:A015709 - cp's OEIS frontend

A015721 Composite and even n such that phi(n) * sigma(n) is one less than a square.

6, 22, 44, 76, 82, 140, 190, 354, 392, 650, 944, 1072, 1142, 1150, 1668, 2276, 4176, 4192, 4262, 5090, 7260, 8660, 15026, 16064, 22926, 29746, 39436, 42872, 74666, 87498, 94786, 101882, 115772, 118928, 127580, 151348

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..300
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Subsequence of even terms of A015709.

isok(n) = !isprime(n) && !(n%2) && issquare(eulerphi(n)*sigma(n)+1); \\ Michel Marcus, Oct 02 2017

Offset corrected by Donovan Johnson, Jan 16 2012

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).

Original entry on oeis.org

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

Amiram Eldar, Aug 11 2019

nonn

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, ...

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

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

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

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

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]

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

cp's OEIS Frontend

A015721 Composite and even n such that phi(n) * sigma(n) is one less than a square.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI