A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2.

25, 70225, 130576327, 189750625, 512706121225, 13837575261123, 99612037019889, 1385331749802025, 3743165875258953025, 10114032809617941274225, 8905398244301708746029223, 27328112908421802064005625, 73840550964522899559001927225

Offset: 1

Author

Jud McCranie, Oct 15 2002

Keywords

• nonn

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

Examples

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

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

Crossrefs

Cf. A001694, A060355.

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