This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A076445 #23 Feb 16 2025 08:32:47 %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 powerful numbers (A001694) 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 (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 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_, May 04 2006 %D A076445 R. K. Guy, Unsolved Problems in Number Theory, B16 %H A076445 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 A076445 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 A076445 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 A076445 Eric Weisstein's World of Mathematics, <a href="https://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, A060355. %K A076445 nonn %O A076445 1,1 %A A076445 _Jud McCranie_, 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_, May 04 2006