A113658 -id:A113658 - cp's OEIS frontend

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

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, 18, 29, 87, 31, 16, 24, 187, 35, 30, 37, 703, 39, 26, 41, 123, 43, 86, 45, 69, 47, 94, 28, 49, 75, 34, 53, 159, 55, 44, 38, 57, 59, 118, 61, 1891, 63, 32, 40, 65, 67, 134, 46, 105

Offset: 1

N. J. A. Sloane, Nov 03 2005

nonn

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.
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
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
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]
From Peter Munn, Jul 27 2019: (Start)
The above conjecture is true.
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.
For even k, k appears by term 2k-1, as k divides T(2k-1) and by induction all smaller divisors have occurred earlier.
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.
(End)
For a generalization of the construction, see A309200. - N. J. A. Sloane, Jul 25 2019
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

Donovan Johnson, Table of n, a(n) for n = 1..10000
Enrique Navarrete, Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Cf. A000217, A111267, A113658 (inverse), A113659 (fixed points), A113702 (trajectory of 10), A309200, A309202, A309203.
For smallest missing numbers see A309195, A309196, A309197.
Indices of squares: A309199.

S:= {}:
for n from 1 to 1000 do
  A111273[n]:= min(numtheory:-divisors(n*(n+1)/2) minus S);
  S:= S union {A111273[n]};
od:
seq(A111273[n],n=1..1000); # Robert Israel, Jan 16 2019

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 *)

{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

def A111273list(upto):
    A = []
    for n in (1..upto):
        D = divisors((n*(n+1)/2))
        A.append(next(d for d in D if d not in A))
    return A
print(A111273list(69)) # Peter Luschny, Jul 26 2019

More terms from Klaus Brockhaus, Nov 03 2005

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

Original entry on oeis.org

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

Klaus Brockhaus, Nov 04 2005

nonn

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.)

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

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.

Cf. A111273, A113658, A016754, A103962.

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 *)

{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,",")))}

A309195 a(n) = smallest number missing from A111273 after A111273(n) has been found.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 6, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 12, 12, 12, 12, 16, 16, 16, 16, 16, 16, 16, 16, 24, 26, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 28, 28, 28, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 36, 36, 36, 36, 36, 36, 36, 36, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 48, 48

Offset: 1

N. J. A. Sloane, Jul 24 2019

nonn

A111273(n) can be even only if the triangular number T_n is even, that is when n is congruent to 0 or 3 modulo 4. So, as A111273(4) is not even, for n >= 4 there is an even number k <= n that has not appeared in A111273 by term n, whereas all odd numbers k <= n have appeared (as explained in A111273). Thus a(n) is even for all n. Also a(n) > n/2 for all n >= 1. - Peter Munn, Jul 27 2019

			1 2 3 4 .5 6 7 8  <- n
1 3 2 5 15 7 4 6  <- A111273
2 2 4 4 .4 4 6 8  <- smallest number missing from A111273 = a(n)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A111273, A113658, A309196, A309197.

N:= 100: # to get a(1)..a(N)
Missing:= {$1..N}:
for n from 1 to N do
  v:= min(numtheory:-divisors(n*(n+1)/2) intersect Missing);
  Missing:= Missing minus {v};
  A[n]:= min(Missing);
od:
seq(A[n],n=1..N); # Robert Israel, Jul 25 2019

The values I gave earlier today were wrong, caused by a bug in my program. Thanks to Peter Munn for pointing out that something was wrong. - N. J. A. Sloane, Jul 24 2019

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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.

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

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A309195 a(n) = smallest number missing from A111273 after A111273(n) has been found.

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A309199 Index of n^2 in A111273.

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A113703 Retrograde trajectory of 10 under map k -> A111273(k).

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