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A076445
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The smaller of a pair of powerful numbers (A001694) that differ by 2.
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17
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25, 70225, 130576327, 189750625, 512706121225, 13837575261123, 99612037019889, 1385331749802025, 3743165875258953025, 10114032809617941274225, 8905398244301708746029223, 27328112908421802064005625, 73840550964522899559001927225
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OFFSET
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1,1
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COMMENTS
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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.
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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B16
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LINKS
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EXAMPLE
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25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005
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STATUS
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approved
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