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A111273 a(n) is the smallest divisor of triangular number T(n) := n(n+1)/2 not already in the sequence.

%I A111273 #60 Oct 03 2019 21:02:18
%S A111273 1,3,2,5,15,7,4,6,9,11,22,13,91,21,8,17,51,19,10,14,33,23,12,20,25,27,
%T A111273 18,29,87,31,16,24,187,35,30,37,703,39,26,41,123,43,86,45,69,47,94,28,
%U A111273 49,75,34,53,159,55,44,38,57,59,118,61,1891,63,32,40,65,67,134,46,105
%N A111273 a(n) is the smallest divisor of triangular number T(n) := n(n+1)/2 not already in the sequence.
%C A111273 A permutation of the natural numbers. Proof: Let k be the smallest number that does not appear. Let n_0 be such that by term n_0 every number < k has appeared. Let m be smallest multiple of k > n_0. Then T(2m) is divisible by k and so a(2m) = k, a contradiction.
%C A111273 Known cycles are: (1), (2, 3), (4, 5, 15, 8, 6, 7), (9), (16, 17, 51, 34, 35, 30, 31), (25) and {28, 29, 87, 58, 59, 118, 119, 68, 46, 47, 94, 95, 48} and the additional fixed-points 49, 57, 65, 81, 85, 93, 121, 133, 153, 169, 185, 201, 209, 217, 225, 253, 261, 289, 297, ... - _John W. Layman_, Nov 09 2005
%C A111273 The trajectory of 10 begins {10, 11, 22, 23, 12, 13, 91, 161, 189, 285, 429, 473, 869, 957, 1437, 2157, 3237, 4857, 7287, 4164, 3470, 4511, 2256, 1464, 1172, 782, 783, 392, 294, 413, 531, 342, 343, 172, 173, 519, 346, 347, 694, 1735, 1388, 926, 927, 464, 248, 166, 167, 84, 70, 71, 36, 37, 703, 352, 353, 1059, 706, 2471, 1412, 1413, 2121, 7427, 6366, 6367, 3184, 1820, 1214, 1215, 608, 336, 337, 4381, 28483, ...) and cannot be further determined without calculating at least the first 28483 terms of {a(n)}. - _John W. Layman_, Nov 09 2005
%C A111273 Conjecture: For all odd primes p, a(p-1) = p.  Equivalently, it appears that if an initial 0 is appended (the smallest divisor of 0, the zeroth triangular number), then the fixed points in this include the odd primes. - _Enrique Navarrete_, Jul 24 2019 [Wording of the equivalent property corrected by _Peter Munn_, Jul 27 2019]
%C A111273 From _Peter Munn_, Jul 27 2019: (Start)
%C A111273 The above conjecture is true.
%C A111273 For odd k, k appears by term k. Proof: choose m such that k-1 <= m <= k and T(m) is odd. k is a divisor of T(m) and (by induction) all smaller odd divisors have occurred earlier, so a(m) = k if k has not occurred earlier.
%C A111273 For even k, k appears by term 2k-1, as k divides T(2k-1) and by induction all smaller divisors have occurred earlier.
%C A111273 For odd prime p, the first triangular number p divides is T(p-1) = p*(p-1)/2. But (p-1)/2 and any smaller divisors have occurred by term (p-1)-1, so a(p-1) = p.
%C A111273 (End)
%C A111273 For a generalization of the construction, see A309200. - _N. J. A. Sloane_, Jul 25 2019
%C A111273 Regarding iteration cycles, for length 2 there are many additional ones after the mentioned (2,3): (50, 75), (122, 183), (174, 203), (194, 291), (338, 507), etc.; for length 3: (1734, 4335, 2312), (4804, 6005, 8407), (7494, 18735, 9992), (8994, 10493, 13491), (12548, 18822, 21959), etc.; for length 4: (84326, 126489, 149487, 91992), (94138, 98417, 135761, 141207), (255206, 382809, 638015, 364580), (345928, 487444, 609305, 680063), (384350, 422785, 499655, 399724), etc. The trajectories of 10 and other families (14, 40, 60, 72, 78, 88, 96, etc.) are best thought of as being continuations of sequences arriving from infinity: ..., 451160, 300774, 300775, 186140, 124094, 124095, 62048, 31304, 25044, 20870, 20871, 13914, 13915, 10934, 10935, 7290, 7291, 14582, 14583, 9722, 9723, 6482, 6483, 4322, 4323, 2882, 4061, 12183, 9138, 9139, 11882, 17823, 8912, 6684, 5570, 5571, 2786, 4179, 2090, 2091, 1394, 1395, 698, 1047, 524, 350, 351, 176, 132, 114, 115, 145, 365, 915, 458, 459, 414, 415, 208, 152, 102, 103, 52, 53, 159, 80, 54, 55, 44, 45, 69, 105, 265, 371, 186, 341, 589, 1121, 1947, 1298, 1299, 866, 867, 578, 579, 290, 435, 218, 219, 146, 147, 74, 111, 56, 38, 39, 26, 27, 18, 19, 10, 11, 22, 23, 12, 13, 91, 161, 189, 285, 429, 473, 869, 957, 1437, 2157, 3237, 4857, 7287, 4164, 3470, 4511, 2256, 1464, 1172, 782, 783, 392, 294, 413, 531, 342, 343, 172, 173, 519, 346, 347, 694, 1735, 1388, 926, 927, 464, 248, 166, 167, 84, 70, 71, 36, 37, 703, 352, 353, 1059, 706, 2471, 1412, 1413, 2121, 7427, 6366, 6367, 3184, 1820, 1214, 1215, 608, 336, 337, 4381, 28483, 49847, 28484, 35605, 89015, 74180, 74181, 101041, 210061, 8297449, ... - _Hans Havermann_, Jul 26 2019
%H A111273 Donovan Johnson, <a href="/A111273/b111273.txt">Table of n, a(n) for n = 1..10000</a>
%H A111273 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.
%p A111273 S:= {}:
%p A111273 for n from 1 to 1000 do
%p A111273   A111273[n]:= min(numtheory:-divisors(n*(n+1)/2) minus S);
%p A111273   S:= S union {A111273[n]};
%p A111273 od:
%p A111273 seq(A111273[n],n=1..1000); # _Robert Israel_, Jan 16 2019
%t A111273 a[n_] := a[n] = Do[If[FreeQ[Array[a, n-1], d], Return[d]], {d, Divisors[n (n+1)/2]}]; Array[a, 100] (* _Jean-François Alcover_, Mar 22 2019 *)
%o A111273 (PARI) {m=69; v=Set([]); for(n=1,m,d=divisors(n*(n+1)/2); j=1; while(setsearch(v,d[j])>0,j++); a=d[j]; v=setunion(v,Set(a)); print1(a,","))} \\ _Klaus Brockhaus_, Nov 03 2005
%o A111273 (Sage)
%o A111273 def A111273list(upto):
%o A111273     A = []
%o A111273     for n in (1..upto):
%o A111273         D = divisors((n*(n+1)/2))
%o A111273         A.append(next(d for d in D if d not in A))
%o A111273     return A
%o A111273 print(A111273list(69)) # _Peter Luschny_, Jul 26 2019
%Y A111273 Cf. A000217, A111267, A113658 (inverse), A113659 (fixed points), A113702 (trajectory of 10), A309200, A309202, A309203.
%Y A111273 For smallest missing numbers see A309195, A309196, A309197.
%Y A111273 Indices of squares: A309199.
%K A111273 nonn
%O A111273 1,2
%A A111273 _N. J. A. Sloane_, Nov 03 2005
%E A111273 More terms from _Klaus Brockhaus_, Nov 03 2005