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

1,1

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

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), 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

Cf. A001694, A060355.

Sequence in context: A337727 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