A227297 Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.

12167, 5425069447, 11968683934831, 28821995554247, 48689748233307

Offset

1,1

Comments

a(1) to a(5) were found by Jaroslaw Wroblewski, who also proved that this sequence is infinite (see link to Problem 53 below). However, there are no more terms less than 500^6 = 1.5625*10^16.

A subsequence of A060355 and of A001694.

References

R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, (1994), pp. 70-74. (See Powerful numbers, section B16.)

Links

Table of n, a(n) for n=1..5.

S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (October 1970), 848-852.

Carlos Rivera, Problem 53: Powerful numbers revisited

David T. Walker, Consecutive integer pairs of powerful numbers and related Diophantine equations, Fibonacci Quart., 14, (1976), pp. 111-116.

Wikipedia, Powerful number

Example

12167 is included in this sequence because (12167, 12168) are a pair of consecutive powerful numbers, neither of which are perfect squares. However, 235224 is not in the sequence because although (235224,235225) are a pair of consecutive powerful numbers, the larger member of the pair is a square number (=485^2).

Crossrefs

Cf. A060355, A001694.

Sequence in context: A167729 A115674 A013819 * A013908 A035915 A104791

Adjacent sequences: A227294 A227295 A227296 * A227298 A227299 A227300

Keyword

nonn,more

Author

Ant King, Jul 07 2013

Status

approved