A309200 -id:A309200 - 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

A309364 a(n) is the least k >= 0 such that n divides C(k) (where C(k) are the Catalan numbers A000108).

Original entry on oeis.org

0, 2, 5, 6, 3, 5, 4, 14, 14, 8, 6, 6, 7, 4, 14, 30, 9, 14, 10, 13, 5, 6, 12, 14, 13, 8, 41, 12, 15, 14, 16, 62, 6, 9, 18, 14, 19, 10, 7, 14, 21, 5, 22, 6, 14, 12, 24, 46, 25, 13, 14, 10, 27, 41, 8, 26, 14, 16, 30, 14, 31, 16, 25, 126, 8, 6, 34, 10, 14, 18, 36

Offset: 1

Rémy Sigrist, Jul 25 2019

nonn

The sequence is well defined:
- if k has t+1 ones in binary representation, 2^t divides C(k),
- for any odd prime number p: if k has e digits (p+1)/2 in base p, p^e divides C(k),
- for any n with prime factorization 2^t * Product_{i=1..o} p_i ^ e_i (where p_i are distinct odd prime numbers),
- by the Chinese remainder theorem, there is a number N ending with t+1 ones in base 2 and ending with e_i digits (p_i+1)/2 in base p_i for i = 1..o,
- C(N) is a multiple of n, and
- a(n) <= N.
As a consequence, A309200 is a permutation of the positive integers (since for any n > 0, we have infinitely many multiples of n among the Catalan number, and then the argument used to prove that A111273 is a permutation completes the proof).

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

Cf. A000108, A309200.

a(n) = for (k=0, oo, my (c=binomial(2*k, k)/(k+1)); if (c%n==0, return (k)))

from itertools import count
def A309364(n):
    if n == 1: return 0
    c = 1
    for k in count(1):
        if not c%n: return k
        c = c*((k<<1)+1<<1)//(k+2) # Chai Wah Wu, May 04 2023

a(p) = (p+1)/2 for any prime number p > 3.

a(C(k)) = k for k <> 1.

A307547 a(n) is the smallest divisor of the partition number P(n) = A000041(n) not already in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 6, 14, 4, 77, 101, 9, 8, 21, 27, 35, 10, 19, 12, 167, 251, 25, 89, 28, 43, 13, 55, 467, 311, 23, 49, 1231, 33, 17977, 281, 121, 45, 42, 193, 2417, 71, 31, 41, 73, 38, 7013, 275, 9283, 363, 53, 63, 17, 142, 47, 102359, 20, 44, 139

Offset: 1

Rémy Sigrist, Jul 27 2019

nonn

Provided A046641(m) is defined for any number m > 0, this sequence is a permutation of the natural numbers.

			The first terms, alongside the divisors of P(n), are:
  n   a(n)  div(P(n))
  --  ----  --------------------
   1     1  (1)
   2     2  (1, 2)
   3     3  (1, 3)
   4     5  (1, 5)
   5     7  (1, 7)
   6    11  (1, 11)
   7    15  (1, 3, 5, 15)
   8    22  (1, 2, 11, 22)
   9     6  (1, 2, 3, 5, 6, 10, 15, 30)
  10    14  (1, 2, 3, 6, 7, 14, 21, 42)

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, PARI program for A307547

Cf. A000041, A046641, A049575, A309200.

PARI
```
See Links section.
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a(A049575(k)) = A049575(k).

A309201 a(n) is the smallest divisor of the Motzkin number A001006(n) not already in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 7, 17, 127, 19, 5, 547, 13, 15511, 15, 6, 9, 284489, 57, 1089397, 12, 73, 11, 21, 35, 63, 119, 6417454619, 38, 107, 31, 1483, 497461, 4644523115569, 51, 10, 37, 953467954114363, 1601, 370537, 1063, 1301337253214147, 43, 18, 1951, 520497658389713341

Offset: 1

N. J. A. Sloane, Jul 25 2019

nonn

Is this a permutation of the positive integers? Daniel Suteu's b-file suggests the answer is no, since powers of 2 >= 8 seem to be missing.
In fact Daniel Suteu points out that Eu and Liu (2008) prove that no Motzkin number is a multiple of 8.
Given any monotonically increasing sequence {b(n): n >= 1} of positive integers we can define a sequence {a(n): n >= 1} by setting a(n) to be smallest divisor of b(n) not already in the {a(n)} sequence. The triangular numbers A000217 produce A111273. A000027 is fixed under this transformation.

Daniel Suteu, Table of n, a(n) for n = 1..191
Eu, Sen-Peng & Liu, Shu-Chung & Yeh, Yeong-nan, Catalan and Motzkin numbers modulo 4 and 8, EJC (2008).

Cf. A000027, A000108, A000217, A111273, A309200.

More terms from Daniel Suteu, Jul 25 2019

A309373 a(n) is the smallest divisor of F(n+1) not already in the sequence (where F denotes the Fibonacci numbers A000045).

Original entry on oeis.org

1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 10, 21, 1597, 8, 37, 15, 26, 199, 28657, 9, 25, 521, 34, 39, 514229, 20, 557, 47, 178, 3571, 65, 12, 73, 113, 466, 33, 2789, 52, 433494437, 43, 61, 139, 2971215073, 14, 97, 55, 3194, 699, 953, 19, 445, 49, 74, 59

Offset: 1

Rémy Sigrist, Jul 26 2019

nonn

This sequence is the Fibonacci variant of A309200.

This sequence is a permutation of the natural numbers as for any m > 0, there are infinitely many multiples of m in A000045.

			The first terms, alongside the divisors of F(n+1), are:
  n   a(n)  div(F(n+1))
  --  ----  ----------------------------------------
   1     1  (1)
   2     2  (1, 2)
   3     3  (1, 3)
   4     5  (1, 5)
   5     4  (1, 2, 4, 8)
   6    13  (1, 13)
   7     7  (1, 3, 7, 21)
   8    17  (1, 2, 17, 34)
   9    11  (1, 5, 11, 55)
  10    89  (1, 89)
  11     6  (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144)
  12   233  (1, 233)

Daniel Suteu, Table of n, a(n) for n = 1..1258
Index entries for sequences that are permutations of the natural numbers
Rémy Sigrist, PARI program for A309373

Cf. A000045, A001177, A309200.

PARI
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See Links section.
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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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A309364 a(n) is the least k >= 0 such that n divides C(k) (where C(k) are the Catalan numbers A000108).

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A307547 a(n) is the smallest divisor of the partition number P(n) = A000041(n) not already in the sequence.

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A309201 a(n) is the smallest divisor of the Motzkin number A001006(n) not already in the sequence.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A309373 a(n) is the smallest divisor of F(n+1) not already in the sequence (where F denotes the Fibonacci numbers A000045).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI