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)).
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
Keywords
Examples
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).
Programs
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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 *)
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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 = ", ")
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