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A060355
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Numbers k such that k and k+1 are powerful numbers.
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22
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8, 288, 675, 9800, 12167, 235224, 332928, 465124, 1825200, 11309768, 384199200, 592192224, 4931691075, 5425069447, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 11968683934831, 13325427460800, 15061377048200, 28821995554247
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OFFSET
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1,1
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COMMENTS
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"Erdős conjectured in 1975 that there do not exist three consecutive powerful integers." - Guy
See Guy for Erdős's conjecture and statement that this sequence is infinite. - Jud McCranie, Oct 13 2002
It is easy to see that this sequence is infinite: if k is in the sequence, so is 4*k*(k+1). - Franklin T. Adams-Watters, Sep 16 2009
The first of a run of three consecutive powerful numbers (conjectured to be empty) are just those in this sequence and A076445. - Charles R Greathouse IV, Nov 16 2012
Jaroslaw Wroblewski (see Prime Puzzles link) shows that there are infinitely many terms k in this sequence such that neither k nor k+1 is a square. - Charles R Greathouse IV, Nov 19 2012
Paul Erdős wrote of meeting Kurt Mahler in 1936: "I almost immediately posed him the following problem: ... are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes! x^2 - 8y^2 = 1 has infinitely many solutions. I was a bit crestfallen since I felt that I should have thought of this myself." - Jonathan Sondow, Feb 08 2015
Of the first 39 terms k, only 7 are such that neither k nor k+1 is a square. - Jon E. Schoenfield, Jun 12 2024
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 288, pp. 74, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B16.
P. Shiu, On the number of square-full integers between successive squares, Volume 27, Issue 2 (December 1980), pp. 171-178.
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LINKS
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EXAMPLE
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1825200 belongs to this sequence because both 1825200 = 2^4 * 3^3 * 5^2 * 13^2 and 1825201 = 7^2 * 193^2 = 1351^2 are powerful numbers. - Labos Elemer, May 03 2001
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MATHEMATICA
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SequencePosition[Table[If[Min[FactorInteger[n][[;; , 2]]]>1, 1, 0], {n, 11310000}], {1, 1}][[;; , 1]] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Mar 27 2024 *)
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PROG
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(Haskell)
a060355 n = a060355_list !! (n-1)
a060355_list = map a001694 $ filter ((== 1) . a076446) [1..]
(Sage)
return a.is_powerful(n) and a.is_powerful(n+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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