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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.

%I A227297 #28 Mar 22 2025 04:41:22
%S A227297 12167,5425069447,11968683934831,28821995554247,48689748233307,
%T A227297 161461422688535037152,3887785221910670811499
%N 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.
%C A227297 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 A227297 A subsequence of A060355 and of A001694.
%D A227297 Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, (1994), pp. 70-74. (See Powerful numbers, section B16.)
%H A227297 Solomon W. Golomb, <a href="http://www.jstor.org/stable/2317020">Powerful numbers</a>, Amer. Math. Monthly, Vol. 77, No. 8 (October 1970), 848-852.
%H A227297 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_053.htm">Problem 53: Powerful numbers revisited</a>, The Prime Puzzles & Problems Connection.
%H A227297 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., Vol. 14, No. 2 (1976), pp. 111-116.
%H A227297 Wikipedia, <a href="http://en.wikipedia.org/wiki/Powerful_number">Powerful number</a>.
%H A227297 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%e A227297 12167 is a term because (12167, 12168) are a pair of consecutive powerful numbers, neither of which are perfect squares.
%e A227297 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).
%Y A227297 Cf. A060355, A001694.
%K A227297 nonn,more
%O A227297 1,1
%A A227297 _Ant King_, Jul 07 2013
%E A227297 a(6)-a(7) from the b-file at A060355 added by _Amiram Eldar_, Mar 22 2025