login
A364581
Numbers k such that the number of iterations of psi(phi(x)) starting at x = k and terminating when psi(phi(x)) = x (k is counted), -1 otherwise is the same for phi(psi(k)).
0
1, 4, 8, 14, 15, 16, 21, 22, 26, 28, 32, 39, 44, 45, 46, 50, 51, 52, 56, 58, 64, 74, 82, 85, 86, 88, 92, 94, 98, 100, 104, 105, 111, 112, 114, 116, 118, 122, 128, 129, 135, 142, 146, 147, 148, 153, 154, 159, 164, 165, 166, 172, 176, 178, 182, 183, 184, 186, 188
OFFSET
1,2
COMMENTS
Numbers k such that A364631(k) = A364642(k).
Conjecture: For each a(n), n > 1, a(n)*7 is a term.
Conjecture: For each even a(n), a(n)*2 is a term.
EXAMPLE
a(1) = 1 is a term because A364631(1) = A364642(1).
a(2) = 4 is a term because A364631(4) = A364642(4).
a(3) = 8 is a term because A364631(8) = A364642(8).
MATHEMATICA
psi[n_] := n*Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Select[Range[200], Length@ FixedPointList[EulerPhi[psi[#1]] &, #] == Length@ FixedPointList[psi[EulerPhi[#1]] &, #] &] (* Amiram Eldar, Aug 04 2023 *)
PROG
(Python)
from sympy.ntheory.factor_ import totient
from sympy import isprime, primefactors, prod
def psi(n):
plist = primefactors(n)
return n*prod(p+1 for p in plist)//prod(plist)
def a364631(n):
i = 1
r = n
while (True):
rc = totient(psi(r))
if (rc == r):
break;
r = rc
i += 1
return i
def a364642(n):
i = 1
r = n
while (True):
rc = psi(totient(r))
if (rc == r):
break;
r = rc
i += 1
return i
# Output display terms.
for n in range(1, 222):
if(a364631(n) == a364642(n)):
print(n, end = ", ")
CROSSREFS
KEYWORD
nonn
AUTHOR
Torlach Rush, Jul 28 2023
STATUS
approved