

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^2my^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

Table of n, a(n) for n=1..13.
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), 801806.
W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272274.
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



