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A060355
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Numbers n such that n and n+1 are a pair of consecutive powerful numbers.
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10
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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' conjecture and statement that this sequence is infinite. - Jud McCranie, Oct 13 2002
It is easy to see that this sequence is infinite: if n is in the sequence, so is 4*n*(n+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 in this sequence such that neither a(n) nor a(n+1) is a square. - Charles R Greathouse IV, Nov 19 2012
Paul Erdős wrote of meeting Kurt Mahler: "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
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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
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LINKS
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Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
C. K. Caldwell, Powerful Numbers
P. Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral. Math. Soc. Gaz., 16 (1) (1989), 1-2.
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2018 (in French).
J. J. O'Connor and E. F. Robertson, Biography of Kurt Mahler
C. Rivera, Problem 53. Powerful numbers revisited, Prime Puzzles
Eric Weisstein's World of Mathematics, Powerful numbers
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EXAMPLE
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1825200 belongs to the sequence because 1825200 = 2*2*2*2*3*3*3*5*5*13*13, 1825201 = 7*7*193*193 = 1351^2, and both are powerful numbers. - Labos Elemer, May 03 2001
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MATHEMATICA
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f[n_]:=First[Union[Last/@FactorInteger[n]]]; Select[Range[2000000], f[#]>1&&f[#+1]>1&] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
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PROG
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(PARI) is(n)=ispowerful(n)&&ispowerful(n+1) \\ Charles R Greathouse IV, Nov 16 2012
(Haskell)
a060355 n = a060355_list !! (n-1)
a060355_list = map a001694 $ filter ((== 1) . a076446) [1..]
-- Reinhard Zumkeller, Jun 03 2015, Nov 30 2012
(Sage)
def A060355(n):
a = sloane.A001694
return a.is_powerful(n) and a.is_powerful(n+1)
[n for n in (1..333333) if A060355(n)] # Peter Luschny, Feb 08 2015
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CROSSREFS
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Primitive elements are in A199801.
Cf. A001694, A060859.
Cf. A076446 (first differences of A001694).
Sequence in context: A136364 A089670 A221612 * A060859 A187289 A187191
Adjacent sequences: A060352 A060353 A060354 * A060356 A060357 A060358
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KEYWORD
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nonn
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AUTHOR
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Jason Earls, Apr 01 2001
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EXTENSIONS
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Corrected and extended by Jud McCranie, Jul 08 2001
More terms from Jud McCranie, Oct 13 2002
a(22)-a(23) from Donovan Johnson, Jul 29 2011
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STATUS
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approved
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