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%I #24 Dec 07 2021 17:17:02
%S 1,2,8,9,1520,1521,50624,50625,62000,62001,103040,103041,199808,
%T 199809,221840,221841,269360,269361,463760,463761,690560,690561,
%U 848240,848241,986048,986049,1252160,1252161,1418480,1418481,2169728,2169729,2692880
%N List of pairs of consecutive refactorable numbers.
%C Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. As a result, a(n) == n-1 (mod 2) for n >= 3. See also A114617. - _Jianing Song_, Apr 01 2021
%H Donovan Johnson, <a href="/A036898/b036898.txt">Table of n, a(n) for n = 1..2000</a>
%H S. Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999, #2.
%H S. Colton, <a href="http://web.archive.org/web/20070831060523/http://www.dai.ed.ac.uk/homes/simonco/research/hr/">HR - Automatic Theory Formation in Pure Mathematics</a>
%H Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
%e 8 is refactorable because tau(8)=4 and 4 divides 8.
%e 9 is refactorable because tau(9)=3 and 3 divides 9.
%t SequencePosition[Table[If[Divisible[n,DivisorSigma[0,n]],1,0],{n,27*10^5}],{1,1}]//Flatten (* _Harvey P. Dale_, Dec 07 2021 *)
%o (PARI) isrefac(n) = ! (n % numdiv(n));
%o lista(nn) = {for (n = 1, nn, if (isrefac(n) && isrefac(n+1), print1(n, ", ", n+1, ", ")););} \\ _Michel Marcus_, Aug 31 2013
%Y Cf. A033950, A114617.
%K nonn
%O 1,2
%A Simon Colton (simonco(AT)cs.york.ac.uk)
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