A113659 -id:A113659 - cp's OEIS frontend

A103962 Odd squares not in A113659.

1521, 4225, 4489, 11449, 15129, 32041, 38025, 67081, 99225, 114921, 136161, 137641, 152881, 173889, 182329, 218089, 221841, 227529, 292681, 294849, 297025, 299209, 301401, 303601, 305809, 308025, 310249, 312481, 314721

Offset: 1

Klaus Brockhaus, Nov 12 2005

nonn

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

Original entry on oeis.org

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

A113658 Where n appears in A111273.

Original entry on oeis.org

1, 3, 2, 7, 4, 8, 6, 15, 9, 19, 10, 23, 12, 20, 5, 31, 16, 27, 18, 24, 14, 11, 22, 32, 25, 39, 26, 48, 28, 35, 30, 63, 21, 51, 34, 71, 36, 56, 38, 64, 40, 83, 42, 55, 44, 68, 46, 95, 49, 75, 17, 103, 52, 80, 54, 111, 57, 87, 58, 104, 60, 92, 62, 127, 65, 99, 66, 119, 45, 84, 70

Offset: 1

Klaus Brockhaus, Nov 03 2005

nonn

Inverse of the permutation of the natural numbers given in A111273.

			A111273(15) = 8, hence a(8) = 15.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A111273, A113659.

A113700 Members of 2-cycles of permutation A111273.

Original entry on oeis.org

2, 3, 50, 75, 122, 174, 183, 194, 203, 291, 338, 410, 507, 615, 722, 794, 842, 914, 1058, 1082, 1083, 1154, 1182, 1191, 1202, 1263, 1346, 1371, 1379, 1442, 1562, 1572, 1587, 1623, 1682, 1703, 1731, 1778, 1803, 1922, 2018, 2019, 2066, 2163, 2343, 2354, 2426

Offset: 1

Klaus Brockhaus, Nov 08 2005

nonn

Trajectory of n under map k -> A111273(k) is periodic with period length 2.
n = A111273(A111273(n)) and n <> A111273(n).
Apparently A111273 has infinitely many 2-cycles.

			A111273(122) = 183 and A111273(183) = 122, hence 122 and 183 are in the
sequence.

Cf. A111273, A113659, A113701.

A113701 Members of 3-cycles of permutation A111273.

Original entry on oeis.org

1734, 2312, 4335, 4804, 6005, 7494, 8407, 8994, 9992, 10493, 12548, 13491, 16004, 18244, 18735, 18822, 19268, 20005, 21956, 21959, 22805, 23412, 24026, 24964, 25363, 26076, 27332, 28007, 28902, 30020, 30692, 31205, 31927, 32934, 33167

Offset: 1

Klaus Brockhaus, Nov 08 2005

nonn

Trajectory of n under map k -> A111273(k) is periodic with period length 3.
n = A111273(A111273(A111273(n))), n <> A111273(A111273(n)), n <> A111273(n).
Apparently A111273 has infinitely many 3-cycles.
The only k-cycles with k > 3 and terms < 240000 are the 4-cycles (84326,126489,149487,91992) and (94138,98417,135761,141207), the 6-cycle (4,5,15,8,6,7), the 7-cycle (16,17,51,34,35,30,31) and the 13-cycle (28,29,87,58,59,118,119,68,46,47,94,95,48).

			A111273(7494) = 18735, A111273(18735) = 9992 and A111273(9992) = 7494, hence
7494, 9992 and 18735 are in the sequence.

Cf. A111273, A113659, A113700.

A113702 Trajectory of 10 under map k -> A111273(k).

Original entry on oeis.org

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

Offset: 0

Klaus Brockhaus, Nov 08 2005

nonn

10 is the smallest number that is not member of a k-cycle with k <=13 of permutation A111273.
Conjecture: Sequence is not periodic.
For the retrograde trajectory of 10 see A113703.

Robert Israel, Table of n, a(n) for n = 0..81

Cf. A111273, A113659, A113700, A113701, A113703.

# assuming A111273 is a list, Vector or table
a113702[0]:= 10:
for i from 1 do
  t:= traperror(A111273[a113702[i-1]]);
if not t::integer then break fi;
  a113702[i]:=t
od:
seq(a113702[j],j=0..i-1); # Robert Israel, Jan 16 2019

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

Original entry on oeis.org

10, 19, 18, 27, 26, 39, 38, 56, 111, 74, 147, 146, 219, 218, 435, 290, 579, 578, 867, 866, 1299, 1298, 1947, 1121, 589, 341, 186, 371, 265, 105, 69, 45, 44, 55, 54, 80, 159, 53, 52, 103, 102, 152, 208, 415, 414, 459, 458, 915, 365, 145, 115, 114, 132, 176, 351, 350

Offset: 0

Klaus Brockhaus, Nov 08 2005

nonn

Also trajectory of 10 under map k -> A113658(k).
Conjecture: Sequence is not periodic.
For the trajectory of 10 under map k -> A111273(k) see A113702.

Cf. A111273, A113658, A113659, A113700, A113701, A113702.

cp's OEIS Frontend

A103962 Odd squares not in A113659.

Views

Author

Keywords

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Extensions

A113658 Where n appears in A111273.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A113700 Members of 2-cycles of permutation A111273.

Views

Author

Keywords

Comments

Examples

Crossrefs

A113701 Members of 3-cycles of permutation A111273.

Views

Author

Keywords

Comments

Examples

Crossrefs

A113702 Trajectory of 10 under map k -> A111273(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Crossrefs