A111273 -id:A111273 - cp's OEIS frontend

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

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,",")))}

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.

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

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

A309197 List of numbers k such that A111273(k) reaches the smallest missing number.

Original entry on oeis.org

1, 3, 7, 8, 15, 19, 23, 31, 32, 39, 48, 63, 71, 83, 95, 103, 111, 127, 143, 147, 151, 159, 167, 175, 195, 199, 207, 211, 215, 223, 224, 255, 271, 279, 287, 319, 327, 343, 351, 359, 367, 371, 383, 391, 399, 415, 431, 435, 447, 463, 464, 511, 543, 559, 579, 583, 595, 607, 639, 655, 663, 687, 703

Offset: 1

N. J. A. Sloane, Jul 24 2019

nonn

			After we reach A111273(8) = 6, the smallest missing number in A111273 is 8 (see A309195). We do not see 8 in A111273 until we reach A111273(15) = 8, so 15 is a term.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A309197

Cf. A111273, A309195, A309196.

PARI
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See Links section.
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A113733 Where records occur in A111273.

Original entry on oeis.org

1, 2, 4, 5, 11, 13, 33, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917

Offset: 1

Klaus Brockhaus, Nov 08 2005

nonn

Cf. A111273, A113732.

A309199 Index of n^2 in A111273.

Original entry on oeis.org

1, 7, 9, 31, 25, 71, 49, 127, 81, 199, 121, 287, 169, 391, 225, 511, 289, 567, 361, 799, 441, 847, 529, 1151, 625, 1351, 729, 1567, 841, 1224, 961, 2047, 1089, 2311, 1225, 2591, 1369, 2887, 845, 3199, 1681, 3087, 1849, 3871, 2025, 3703, 2209, 4607, 2401, 4375

Offset: 1

N. J. A. Sloane, Jul 25 2019

nonn

A111273(a(n)) = n^2.

a(n) = A113658(n^2). - Rémy Sigrist, Jul 25 2019

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A309199

Cf. A111273, A113658.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Jul 25 2019

A309203 Numbers k such that A111273(k-1) = k.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 53, 55, 59, 61, 63, 67, 71, 73, 77, 79, 83, 89, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 135, 137, 139, 141, 147, 149, 151, 155, 157, 163, 167, 171, 173, 179, 181, 191, 193, 197, 199, 207, 211, 215, 219

Offset: 1

N. J. A. Sloane, Jul 27 2019

nonn

All terms are odd; includes all odd primes.

The odd terms that are not primes are 27, 35, 39, 45, 55, 63, 77, 95, 99, 115, 119, 125, 135, 141, 147, 155, 171, .... What are these numbers?

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

Cf. A111273.

S:= {}:
R:= NULL: count:= 0:
for n from 1 while count < 200 do
  v:= min(numtheory:-divisors(n*(n+1)/2) minus S);
  S:= S union {v};
  if v = n+1 then R:= R, n+1; count:= count+1 fi;
od:
R; # Robert Israel, Jul 30 2019

cp's OEIS Frontend

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

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A113658 Where n appears in A111273.

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A113700 Members of 2-cycles of permutation A111273.

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

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A113701 Members of 3-cycles of permutation A111273.

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

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A309197 List of numbers k such that A111273(k) reaches the smallest missing number.

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A113733 Where records occur in A111273.

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A309199 Index of n^2 in A111273.

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A309203 Numbers k such that A111273(k-1) = k.

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