1,1

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 (noe(AT)sspectra.com), May 04 2006

R. K. Guy, Unsolved Problems in Number Theory, B16

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.

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.]

Eric Weisstein's World of Mathematics, Powerful numbers

25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence.

Cf. A001694.

Sequence in context: A082211 A053766 A034711 this_sequence A013835 A068737 A151649

Adjacent sequences: A076442 A076443 A076444 this_sequence A076446 A076447 A076448

nonn

Jud McCranie (JudMcCr(AT)BellSouth.net), Oct 15 2002

a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005

More terms from T. D. Noe (noe(AT)sspectra.com), May 04 2006