A389593 Numbers k such that sigma(k)^2 = psi(k)^2 + phi(k)^2.

99, 4743, 8307, 331731, 364941, 497781, 796671, 1081251, 1244709, 2525211, 2641509, 2824263, 4768101, 6479343, 22129731, 43777053, 55722717, 61301187, 65680893, 75623967, 77616963, 85387293, 94282389, 111207969, 116468613, 122622453, 125741061, 133821297, 137638233

Offset

1,1

Links

Table of n, a(n) for n=1..29.

Example

99 is in the sequence since sigma(99)^2 = 156^2 = 144^2 + 60^2 = psi(99)^2 + phi(99)^2.

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1);
f2[p_, e_] := (p + 1)*p^(e - 1);
f3[p_, e_] := (p - 1)*p^(e - 1);
q[k_] := Module[{f = FactorInteger[k]}, (Times @@ f1 @@@ f)^2 == (Times @@ f2 @@@ f)^2 + (Times @@ f3 @@@ f)^2]; Select[Range[2, 10^6], q] (* Amiram Eldar, Oct 08 2025 *)

Prog

(PARI) isok(k) = my(f = factor(k)); sigma(f)^2 == prod(k=1, #f~, (f[k, 1]+1)*f[k, 1]^(f[k, 2]-1))^2 + eulerphi(f)^2; \\ Amiram Eldar, Oct 08 2025

Crossrefs

Cf. A000010, A000203, A001615, A389478.

Sequence in context: A163040 A133319 A196745 * A196903 A017815 A017762

Adjacent sequences: A389590 A389591 A389592 * A389594 A389595 A389596

Keyword

nonn

Author

S. I. Dimitrov, Oct 08 2025

Extensions

More terms from Amiram Eldar, Oct 08 2025

Status

approved