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%I #38 Jun 01 2026 10:14:18
%S 130576327,8905398244301708746029223
%N Powerful numbers of the form k^2 - 2.
%C Note that k^2 - 4k + 4, k^2 - 2k + 1, and k^2 - 2 are powerful numbers in arithmetic progression. van Doorn proves that this sequence is infinite (by constructing an explicit infinite subsequence) and conjectures that, infinitely often, the three terms are consecutive as powerful numbers, which would disprove a conjecture of Erdős.
%H Wouter van Doorn, <a href="https://arxiv.org/abs/2605.06697">Three-term arithmetic progressions of consecutive powerful numbers</a>, arXiv preprint (2026). arXiv:2605.06697 [math.NT]
%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F a(n) < 68200709735854334^n.
%Y Subsequence of A076445 (and hence A001694).
%K nonn,bref
%O 1,1
%A _Charles R Greathouse IV_, May 21 2026