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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.
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 (zevi_35711(AT)yahoo.com), 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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