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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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 *)

Crossrefs

Cf. A127652, A097032, A097010, A098185, A127654, A063991, A097037, A097036.

Sequence in context: A206286 A116023 A062408 * A050705 A095406 A337028

Adjacent sequences: A127650 A127651 A127652 * A127654 A127655 A127656

Keyword

nonn

Author

Ant King, Jan 24 2007

Extensions

More terms from Amiram Eldar, Mar 11 2023

Status

approved