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 (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
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
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