A391967 Numbers k such that sigma(k) + phi(k) = psi(k) + k.

1, 672, 3120, 7800, 11904, 23712, 55552, 195072, 302784, 492864, 910336, 1607808, 1671072, 1750656, 6688560, 13210368, 15242496, 16125952, 16721400, 17033984, 17274624, 26854656, 56755968, 633404992, 805208064, 874527872, 904786944, 1524577280, 2059528512, 2226866176

Offset

1,2

Links

Table of n, a(n) for n=1..30.

Example

672 is a term since sigma(672) + phi(672) = 2016 + 192 = 2208, and psi(672) + 672 = 1536 + 672 = 2208.

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[1] = True; q[k_] := Module[{fct = FactorInteger[k]}, Times @@ f1 @@@ fct + Times @@ f2 @@@ fct == Times @@ f3 @@@ fct + k]; Select[Range[10^6], q] (* Amiram Eldar, Dec 24 2025 *)

Prog

(PARI) isok(n) = { my(f = factor(n)); sigma(f) + eulerphi(f) == n * prod(i=1, #f~, 1 + 1/f[i, 1]) + n; }
(Python)
from math import prod
from sympy import factorint
def ok(n):
    f = factorint(n)
    sigma = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
    phi = prod((p-1)*p**(e-1) for p, e in f.items())
    psi = prod((p+1)*p**(e-1) for p, e in f.items())
    return sigma + phi == psi + n
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Dec 24 2025

Crossrefs

Cf. A000203 (sigma), A000010 (phi), A001615 (psi).

Sequence in context: A057797 A057802 A231740 * A245778 A245786 A321116

Adjacent sequences: A391964 A391965 A391966 * A391968 A391969 A391970

Keyword

nonn

Author

Kaloian Ivanov, Dec 24 2025

Extensions

a(22)-a(30) from Michael S. Branicky, Dec 25 2025

Status

approved