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A127654
Unitary aspiring numbers.
6
66, 78, 244, 292, 476, 482, 578, 648, 680, 688, 770, 784, 832, 864, 956, 958, 976, 1168, 1354, 1360, 1392, 1488, 1600, 1658, 1670, 1906, 2232, 2264, 2294, 2376, 2480, 2552, 2572, 2576, 2626, 2712, 2732, 2806, 2842, 2870, 2904, 2912, 2992, 3024, 3096, 3140, 3172
OFFSET
1,1
COMMENTS
A unitary aspiring number is an integer whose unitary aliquot sequences ends by meeting a unitary-perfect number (A098185) in its trajectory, but is not unitary-perfect itself. There are 1693 such numbers <=100000 and of these 82860 and 97020 generate the longest unitary aliquot sequences (according to A097032), each having length 18 and ending with the unitary perfect number 90.
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) = 476 because the fifth non-unitary-perfect number whose unitary aliquot sequence ends in a unitary-perfect number is 476.
MATHEMATICA
UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #, n/# ] == 1 \ &]; sstar[n_] := Plus @@ UnitaryDivisors[ n] - n; g[n_] := If[n > 0, sstar[n], 0]; UnitaryTrajectory[n_] := Most[NestWhileList[ g, n, UnsameQ, All]]; UnitaryPerfectNumberQ[0] = 0; UnitaryPerfectNumberQ[k_] := If[sstar[k] == k, True, False]; UnitaryAspiringNumberQ[k_] := If[UnitaryPerfectNumberQ[Last[ UnitaryTrajectory[k]]] && ! UnitaryPerfectNumberQ[k], True, False]; Select[Range[2500], UnitaryAspiringNumberQ[ # ] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[3200], 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