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Search: id:A117594
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| A117594 |
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Numbers whose fifth powers are closer to cubic numbers than square numbers. |
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1
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| 199, 1354, 4995, 7320, 7994, 12634, 44217, 91116, 177682, 394826, 458908, 462763, 512012, 1706886, 1738064, 1801677, 1880465, 2523441, 5691648, 6714911, 8383950, 8403388, 11100341, 14706104, 14706146, 15460136, 16337238, 18898872, 21194961
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Numbers which are cubes themselves are excluded as trivial.
It appears that this sequence is infinite. For seventh powers < 10^49, only 2^7 and 3^7 are closer to cubes than squares. Note that 1/2+1/3+1/5>1, but 1/2+1/3+1/7<1. Do these inequalities determine whether there are an infinite or finite number of solutions? Mazur discusses how the ABC conjecture applies to perfect power problems. - T. D. Noe (noe(AT)sspectra.com), Apr 07 2006
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LINKS
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B. Mazur, Questions about Number
Eric Weisstein's World of Mathematics, MathWorld: Perfect Power
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EXAMPLE
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The distance of 199^5 to the nearest cube is 49688. To the nearest square is 165882.
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MATHEMATICA
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nMax=10^6; lst={}; Do[n5=n^5; n3=Round[n5^(1/3)]^3; n2=Round[n5^(1/2)]^2; If[0<Abs[n5-n3]<Abs[n5-n2], AppendTo[lst, n]], {n, nMax}]; lst (Noe) - T. D. Noe (noe(AT)sspectra.com), Apr 07 2006
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CROSSREFS
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Cf. A117934 (perfect powers that are close).
Sequence in context: A105975 A095995 A142570 this_sequence A052188 A086977 A069244
Adjacent sequences: A117591 A117592 A117593 this_sequence A117595 A117596 A117597
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KEYWORD
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nonn
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AUTHOR
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Ed Pegg Jr (ed(AT)mathpuzzle.com), Apr 05 2006
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EXTENSIONS
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More terms from T. D. Noe (noe(AT)sspectra.com) and Hans Havermann (pxp(AT)rogers.com), Apr 08 2006
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