A127653 - OEIS

%I #8 Mar 11 2023 08:05:24

%S 1,10,12,14,15,18,20,21,22,24,26,28,33,34,35,36,38,39,40,44,45,46,48,

%T 50,51,52,55,56,57,58,62,63,65,68,69,70,72,74,75,76,77,80,82,84,85,86,

%U 87,88,91,92,93,94,95,96,98,99,104,105,106,108,110,111,112,115

%N 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.

%H Amiram Eldar, <a href="/A127653/b127653.txt">Table of n, a(n) for n = 1..10000</a>

%H Herman J. J. te Riele, <a href="https://ir.cwi.nl/pub/9137">Unitary Aliquot Sequences</a>, MR 139/72, Mathematisch Centrum, Amsterdam, 1972.

%H Herman J. J. te Riele, <a href="https://ir.cwi.nl/pub/9068">Further Results on Unitary Aliquot Sequences</a>, NW 2/73, Mathematisch Centrum, Amsterdam, 1973.

%e 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.

%t 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[ # ] &]

%t 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 *)

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

%K nonn

%O 1,2

%A _Ant King_, Jan 24 2007

%E More terms from _Amiram Eldar_, Mar 11 2023