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, 161461422688535037152, 3887785221910670811499

Offset: 1

Author

Ant King, Jul 07 2013

Keywords

• nonn
• more

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.

Examples

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

References

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

Links

• Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (October 1970), 848-852.
• Carlos Rivera, Problem 53: Powerful numbers revisited, The Prime Puzzles & Problems Connection.
• David T. Walker, Consecutive integer pairs of powerful numbers and related Diophantine equations, Fibonacci Quart., Vol. 14, No. 2 (1976), pp. 111-116.
• Wikipedia, Powerful number.
• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A060355, A001694.

Extensions

a(6)-a(7) from the b-file at A060355 added by Amiram Eldar, Mar 22 2025