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%I #9 Oct 03 2019 21:04:19
%S 1,9,25,49,57,65,81,85,93,121,133,153,169,185,201,209,217,225,253,261,
%T 289,297,301,305,309,329,333,345,361,369,381,385,393,417,441,469,477,
%U 489,497,501,505,513,525,529,533,553,561,565,581,621,625,633,637,645
%N Numbers n such that A111273(n) = n.
%C Fixed-points of the permutation of the natural numbers given in A111273.
%C Most odd squares appear to be in this sequence, e.g. 1,9,25,49,81,121,139,... The smallest odd square not appearing is 39^2=1521. - _John W. Layman_, Nov 10 2005. (See A103962.)
%H Donovan Johnson, <a href="/A113659/b113659.txt">Table of n, a(n) for n = 1..1000</a>
%H Enrique Navarrete, Daniel Orellana, <a href="https://arxiv.org/abs/1907.10023">Finding Prime Numbers as Fixed Points of Sequences</a>, arXiv:1907.10023 [math.NT], 2019.
%e The first nine terms of A111273 are {1,3,2,5,15,7,4,6,9,...}, so 1 and 9 are fixed-points.
%t Select[MapIndexed[{First@ #2, #1} &, Nest[Function[{a, D}, Append[a, SelectFirst[D, FreeQ[a, #] &] /. k_ /; ! IntegerQ@ k -> Nothing]] @@ {#, Divisors@ PolygonalNumber[Length@ # + 1]} &, {1}, 645] ], SameQ @@ # &][[All, 1]] (* _Michael De Vlieger_, Oct 03 2019 *)
%o (PARI) {m=650;v=Set([]);w=[];for(k=1,m,d=divisors(k*(k+1)/2);j=1;while(setsearch(v,d[j])>0,j++);a=d[j];v=setunion(v,Set(a));w=concat(w,a));for(n=1,m,if(n==w[n],print1(n,",")))}
%Y Cf. A111273, A113658, A016754, A103962.
%K nonn
%O 1,2
%A _Klaus Brockhaus_, Nov 04 2005
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