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 A060355 Numbers n such that n and n+1 are a pair of consecutive powerful numbers. 10

%I

%S 8,288,675,9800,12167,235224,332928,465124,1825200,11309768,384199200,

%T 592192224,4931691075,5425069447,13051463048,221322261600,

%U 443365544448,865363202000,8192480787000,11968683934831,13325427460800,15061377048200,28821995554247

%N Numbers n such that n and n+1 are a pair of consecutive powerful numbers.

%C "Erdős conjectured in 1975 that there do not exist three consecutive powerful integers." - Guy

%C See Guy for Erdős' conjecture and statement that this sequence is infinite. - _Jud McCranie_, Oct 13 2002

%C 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

%C 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

%C 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

%C 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

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 288, pp 74, Ellipses, Paris 2008.

%D R. K. Guy, Unsolved Problems in Number Theory, B16

%H Donovan Johnson, <a href="/A060355/b060355.txt">Table of n, a(n) for n = 1..39</a> (terms < 10^22)

%H C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=PowerfulNumber">Powerful Numbers</a>

%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1989-34.pdf">Some personal and mathematical reminiscences of Kurt Mahler</a>, Austral. Math. Soc. Gaz., 16 (1) (1989), 1-2.

%H J. J. O'Connor and E. F. Robertson, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Mahler.html">Biography of Kurt Mahler</a>

%H C. Rivera, <a href="http://www.primepuzzles.net/problems/prob_053.htm">Problem 53. Powerful numbers revisited</a>, Prime Puzzles

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful numbers</a>

%e 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

%t f[n_]:=First[Union[Last/@FactorInteger[n]]];Select[Range[2000000],f[#]>1&&f[#+1]>1&] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)

%o (PARI) is(n)=ispowerful(n)&&ispowerful(n+1) \\ _Charles R Greathouse IV_, Nov 16 2012

%o a060355 n = a060355_list !! (n-1)

%o a060355_list = map a001694 \$ filter ((== 1) . a076446) [1..]

%o -- _Reinhard Zumkeller_, Jun 03 2015, Nov 30 2012

%o (Sage)

%o def A060355(n):

%o a = sloane.A001694

%o return a.is_powerful(n) and a.is_powerful(n+1)

%o [n for n in (1..333333) if A060355(n)] # _Peter Luschny_, Feb 08 2015

%Y Primitive elements are in A199801.

%Y Cf. A001694, A060859.

%Y Cf. A076446 (first differences of A001694).

%K nonn

%O 1,1

%A Jason Earls (zevi_35711(AT)yahoo.com), Apr 01 2001

%E Corrected and extended by _Jud McCranie_, Jul 08 2001

%E More terms from _Jud McCranie_, Oct 13 2002

%E a(22)-a(23) from _Donovan Johnson_, Jul 29 2011

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