Search: id:A076445
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%I A076445
%S A076445 25,70225,130576327,189750625,512706121225,13837575261123,
%T A076445 99612037019889,1385331749802025,3743165875258953025,
%U A076445 10114032809617941274225,8905398244301708746029223,27328112908421802064005625,
               73840550964522899559001927225
%N A076445 The smaller of a pair of consecutive powerful numbers (using definition 
               1) that differ by 2.
%C A076445 Erdos conjectured that there aren't three consecutive powerful numbers 
               and no examples are known. There are an infinite number of powerful 
               numbers differing by 1. A requirement for three consecutive powerful 
               numbers is a pair that differ by 2 (necessarily odd). These pairs 
               are much more rare.
%C A076445 Sentance gives a method for constructing families of these numbers from 
               the solutions of Pell equations x^2-my^2=1 for certain m whose square 
               root has a particularly simple form as a continued fraction. Sentance's 
               result can be generalized to any m such that A002350(m) is even. 
               These m, which generate all consecutive odd powerful numbers, are 
               in A118894. - T. D. Noe (noe(AT)sspectra.com), May 04 2006
%D A076445 R. K. Guy, Unsolved Problems in Number Theory, B16
%D A076445 R. A. Mollin and P. G. Walsh, On powerful numbers, IJMMS 9:4 (1986), 
               801-806.
%D A076445 W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. 
               Math. Monthly, 88 (1981), 272-274.
%H A076445 Max Alekseyev, <a href="a076445.txt">Conjectured table of n, a(n) for 
               n = 1..33</a> [These terms certainly belong to the sequence, but 
               they are not known to be consecutive.]
%H A076445 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PowerfulNumber.html">Powerful numbers</a>
%e A076445 25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the 
               sequence.
%Y A076445 Cf. A001694.
%Y A076445 Sequence in context: A082211 A053766 A034711 this_sequence A013835 A068737 
               A151649
%Y A076445 Adjacent sequences: A076442 A076443 A076444 this_sequence A076446 A076447 
               A076448
%K A076445 nonn
%O A076445 1,1
%A A076445 Jud McCranie (JudMcCr(AT)BellSouth.net), Oct 15 2002
%E A076445 a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005
%E A076445 More terms from T. D. Noe (noe(AT)sspectra.com), May 04 2006

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