A227297 - OEIS

%I #19 Jan 05 2025 19:51:40

%S 12167,5425069447,11968683934831,28821995554247,48689748233307

%N 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.

%C 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.

%C A subsequence of A060355 and of A001694.

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

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2317020">Powerful numbers</a>, Amer. Math. Monthly, Vol. 77 (October 1970), 848-852.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_053.htm">Problem 53: Powerful numbers revisited</a>

%H David T. Walker, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-2/walker.pdf">Consecutive integer pairs of powerful numbers and related Diophantine equations</a>, Fibonacci Quart., 14, (1976), pp. 111-116.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Powerful_number">Powerful number</a>

%e 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).

%Y Cf. A060355, A001694.

%K nonn,more

%O 1,1

%A _Ant King_, Jul 07 2013