%I #22 Jun 23 2022 11:19:02

%S 25,70225,130576327,189750625,512706121225,13837575261123,

%T 99612037019889,1385331749802025,3743165875258953025,

%U 10114032809617941274225,8905398244301708746029223,27328112908421802064005625,73840550964522899559001927225

%N The smaller of a pair of powerful numbers (A001694) that differ by 2.

%C 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 (cf. A060355). A requirement for three consecutive powerful numbers is a pair that differ by 2 (necessarily odd). These pairs are much more rare.

%C 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_, May 04 2006

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

%H Max Alekseyev, <a href="/A076445/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 R. A. Mollin and P. G. Walsh, <a href="http://www.emis.de/journals/HOA/IJMMS/Volume9_4/812820.pdf">On powerful numbers</a>, IJMMS 9:4 (1986), 801-806.

%H W. A. Sentance, <a href="http://www.jstor.org/stable/2320553">Occurrences of consecutive odd powerful numbers</a>, Amer. Math. Monthly, 88 (1981), 272-274.

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

%e 25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence.

%Y Cf. A001694, A060355.

%K nonn

%O 1,1

%A _Jud McCranie_, Oct 15 2002

%E a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005

%E More terms from _T. D. Noe_, May 04 2006