A113659 Numbers n such that A111273(n) = n.

1, 9, 25, 49, 57, 65, 81, 85, 93, 121, 133, 153, 169, 185, 201, 209, 217, 225, 253, 261, 289, 297, 301, 305, 309, 329, 333, 345, 361, 369, 381, 385, 393, 417, 441, 469, 477, 489, 497, 501, 505, 513, 525, 529, 533, 553, 561, 565, 581, 621, 625, 633, 637, 645

Offset: 1

Author

Klaus Brockhaus, Nov 04 2005

Keywords

• nonn

Comments

Fixed-points of the permutation of the natural numbers given in A111273.

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.)

Examples

			The first nine terms of A111273 are {1,3,2,5,15,7,4,6,9,...}, so 1 and 9 are fixed-points.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Enrique Navarrete, Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Crossrefs

Cf. A111273, A113658, A016754, A103962.

Programs

Mathematica

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 *)

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,",")))}