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Search: id:A127654
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| A127654 |
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Unitary aspiring numbers. |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.
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EXAMPLE
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a(5)=476 because the fifth non unitary-perfect number whose unitary aliquot sequence ends in a unitary-perfect number is 476.
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MATHEMATICA
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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[ # ] &]
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CROSSREFS
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Cf. A097032, A127652, A097010, A098185, A127653, A063991, A097037, A097036.
Sequence in context: A121478 A128896 A109750 this_sequence A032485 A031411 A098775
Adjacent sequences: A127651 A127652 A127653 this_sequence A127655 A127656 A127657
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KEYWORD
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hard,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Jan 24 2007
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