This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2. 14
 25, 70225, 130576327, 189750625, 512706121225, 13837575261123, 99612037019889, 1385331749802025, 3743165875258953025, 10114032809617941274225, 8905398244301708746029223, 27328112908421802064005625, 73840550964522899559001927225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, B16 LINKS Max Alekseyev, Conjectured table of n, a(n) for n = 1..33 [These terms certainly belong to the sequence, but they are not known to be consecutive.] R. A. Mollin and P. G. Walsh, On powerful numbers, IJMMS 9:4 (1986), 801-806. W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274. Eric Weisstein's World of Mathematics, Powerful numbers EXAMPLE 25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence. CROSSREFS Cf. A001694. Sequence in context: A203543 A034711 A325215 * A013835 A211600 A068737 Adjacent sequences:  A076442 A076443 A076444 * A076446 A076447 A076448 KEYWORD nonn AUTHOR Jud McCranie, Oct 15 2002 EXTENSIONS a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005 More terms from T. D. Noe, May 04 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)