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A127653
Integers whose unitary aliquot sequences terminate in 0, including 1 but excluding the other trivial cases in which n is itself either a prime or a prime power.
6
1, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 104, 105, 106, 108, 110, 111, 112, 115
OFFSET
1,2
LINKS
Herman J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, Amsterdam, 1972.
Herman J. J. te Riele, Further Results on Unitary Aliquot Sequences, NW 2/73, Mathematisch Centrum, Amsterdam, 1973.
EXAMPLE
a(5) = 15 because the fifth integer that is neither prime nor a prime power and whose unitary aliquot sequence terminates in 0 is 15.
MATHEMATICA
UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #, n/# ] == 1 \ &]; sstar[n_] := Plus @@ UnitaryDivisors[ n] - n; pp[k_] := If[Length[ FactorInteger[k]] == 1, True, False]; g[n_] := If[n > 0, sstar[n], 0]; UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[100], Last[UnitaryTrajectory[ # ]] == 0 && ! pp[ # ] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := If[PrimeNu[n] == 1, False, Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] == 0]]; Select[Range[120], q] (* Amiram Eldar, Mar 11 2023 *)
KEYWORD
nonn
AUTHOR
Ant King, Jan 24 2007
EXTENSIONS
More terms from Amiram Eldar, Mar 11 2023
STATUS
approved