A111273 -id:A111273 - cp's OEIS frontend

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

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.

A113732 Records in A111273.

Original entry on oeis.org

1, 3, 5, 15, 22, 91, 187, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653

Offset: 1

Klaus Brockhaus, Nov 08 2005

nonn

Cf. A111273, A113733.

A309202 Peter Munn showed in A111273 that if A111273(n)=m then if m is odd, n <= m, and if m is even, n <= 2m-1; a(n) is either m-n or 2m-1-n in the two cases.

Original entry on oeis.org

0, 1, 0, 1, 10, 1, 0, 3, 0, 1, 32, 1, 78, 7, 0, 1, 34, 1, 0, 7, 12, 1, 0, 15, 0, 1, 8, 1, 58, 1, 0, 15, 154, 1, 24, 1, 666, 1, 12, 1, 82, 1, 128, 1, 24, 1, 140, 7, 0, 25, 16, 1, 106, 1, 32, 19, 0, 1, 176, 1, 1830, 1, 0, 15, 0, 1, 200, 23, 36, 1, 0, 1, 2628, 37, 24, 1, 66, 1, 236, 27, 0, 1, 0, 55, 0, 43

Offset: 1

N. J. A. Sloane, Jul 26 2019

nonn

It is known that if p is an odd prime, a(p-1) = 1 (see A111273).

Cf. A111273.

A309200 a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence.

Original entry on oeis.org

1, 2, 5, 7, 3, 4, 11, 10, 13, 17, 14, 19, 20, 6, 9, 15, 22, 12, 21, 23, 26, 8, 18, 29, 28, 31, 34, 35, 37, 16, 41, 38, 30, 39, 43, 46, 47, 25, 49, 44, 27, 24, 36, 40, 42, 45, 51, 50, 52, 33, 53, 54, 55, 48, 57, 58, 59, 60, 61, 56, 63, 32, 65, 66, 67, 68, 69

Offset: 1

N. J. A. Sloane, Jul 25 2019

nonn

Conjecture: This is a permutation of the positive integers. [The conjecture is true, see A309364. - Rémy Sigrist, Jul 25 2019]

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.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A309200

Cf. A000027, A000108, A000217, A111273, A309364..

with(numtheory);
# the general transformation
f := proc(b) local t1,d,j,dlis,L,hit,i,n,a,n1;
if whattype(b) <> list then RETURN([]); fi;
n1:=nops(b); a:=[]; L:=10000;
hit:=Array(0..L,0);
for n from 1 to n1 do
   t1:=b[n];
dlis:=sort(convert(divisors(t1),list));
for j from 1 to nops(dlis) do d:=dlis[j];
  if d > L then error("d too large",n,t1,d); fi;
   if hit[d]=0 then break; fi; od:
a:=[op(a),d];
hit[d]:=1;          od;
[seq(a[i],i=1..nops(a))];
end;
# the Catalan numbers
C:=[seq(binomial(2*n,n)/(n+1),n=1..40)];
f(C);

PARI
```
\\ See Links section.
```

def transform(sup, fun):
    A = []
    for n in (1..sup):
        D = divisors(fun(n))
        A.append(next(d for d in D if d not in A))
    return A
A309200list = lambda lim: transform(lim, catalan_number)
print(A309200list(29)) # Peter Luschny, Jul 26 2019

More terms from Rémy Sigrist, Jul 25 2019

A309196 a(n) = A309195(n)/2.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 24, 24

Offset: 1

N. J. A. Sloane, Jul 24 2019

nonn

Peter Munn has shown that A309195(n) is even for n >= 1 (see that entry for a proof), so this sequence is well-defined. - N. J. A. Sloane, Jul 26 2019

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

Cf. A111273, A309195.

The values I gave yesterday 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

A309275 a(n) is the smallest divisor not yet in the sequence of 3T(n)= 3n(n-1)/2, where T(n) are the triangular numbers; n => 1.

Original entry on oeis.org

1, 3, 9, 2, 5, 15, 7, 4, 6, 27, 11, 18, 13, 21, 35, 8, 12, 17, 19, 10, 14, 33, 23, 36, 20, 25, 39, 42, 29, 45, 31, 16, 22, 51, 85, 30, 37, 57, 117, 26, 41, 63, 43, 66, 54, 69, 47, 24, 28, 49, 75, 34, 53, 81, 55, 44, 38, 87, 59, 90, 61, 93, 189, 32, 40, 65, 67, 102

Offset: 1

Enrique Navarrete, Jul 20 2019

nonn

Up to n = 10000, 1160 of the first 1228 odd primes appear as fixed points of a(n).

			For n = 7, 3*T(7) = 63, and a(7) = 7 is a fixed point and the smallest divisor of 63 not yet in the sequence.
For n = 43, 3*T(43) = 2709, and a(43) = 43 is a fixed point and the smallest divisor of 2709 not yet in the sequence.

Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Cf. A000217, A111273.

A309276 a(n) is the smallest divisor not yet in the sequence of 5T(n)= 5n(n-1)/2, where T(n) are the triangular numbers; n => 1.

Original entry on oeis.org

1, 5, 3, 2, 10, 15, 7, 4, 6, 9, 11, 22, 13, 35, 21, 8, 17, 45, 19, 25, 14, 33, 23, 12, 20, 65, 27, 18, 29, 75, 31, 16, 24, 51, 85, 30, 37, 95, 39, 26, 41, 105, 43, 55, 50, 69, 47, 40, 28, 49, 125, 34, 53, 135, 99, 44, 38, 57, 59, 118, 61, 155, 63, 32, 52, 143, 67, 134, 46, 115, 71

Offset: 1

Enrique Navarrete, Jul 20 2019

nonn

Up to n = 10000, 1145 of the first 1228 odd primes appear as fixed points of a(n).

			For n = 1, 5*T(1) = 0, and a(1) = 1 is the smallest divisor of 0 not yet in the sequence.
For n = 3, 5*T(3) = 15, and a(3) = 3 is a fixed point and the smallest divisor of 15 not yet in the sequence.
For n = 71, 5*T(71) = 2485, and a(71) = 71 is a fixed point and the smallest divisor of 2485 not yet in the sequence.

Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Cf. A000217, A111273.

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.

A309387 a(n) is the smallest positive divisor not yet in the sequence of 7*A000217(n-1); n >= 1.

Original entry on oeis.org

1, 7, 3, 2, 5, 15, 21, 4, 6, 9, 11, 14, 13, 49, 35, 8, 17, 51, 19, 10, 30, 33, 23, 12, 20, 25, 27, 18, 29, 87, 31, 16, 22, 77, 85, 42, 37, 133, 39, 26, 28, 41, 43, 86, 45, 63, 47, 24, 56, 175, 75, 34, 53, 159, 55, 44, 38, 57, 59, 70, 61, 217, 93, 32, 40, 65, 67, 119, 46, 69, 71, 36, 73, 259

Offset: 1

Enrique Navarrete, Jul 27 2019

nonn

Up to n=10000, 1166 of the first 1228 odd primes appear as fixed points of a(n), i.e., 95%.

Conjecture: for large p prime, the odd primes (except p) appear as fixed points of b(n), where b(n) is the smallest positive divisor not yet in the sequence of p*A000217(n-1); n >= 1 (see link).

			For n = 1: a(1) = 1 is the smallest divisor of 7*0 not yet in the sequence.
For n = 23: a(23) = 23 is a fixed point and the smallest divisor of 7*253 not yet in the sequence.
For n = 73: a(73) = 73 is a fixed point and the smallest divisor of 7*2628 not yet in the sequence.

Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.

Cf. A000217, A111273, A309275, A309276.

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

cp's OEIS Frontend

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

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A113732 Records in A111273.

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A309202 Peter Munn showed in A111273 that if A111273(n)=m then if m is odd, n <= m, and if m is even, n <= 2*m-1; a(n) is either m-n or 2*m-1-n in the two cases.

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A309200 a(n) is the smallest divisor of the Catalan number C(n) = A000108(n) not already in the sequence.

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Maple

PARI

Sage

Extensions

A309196 a(n) = A309195(n)/2.

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Extensions

A309275 a(n) is the smallest divisor not yet in the sequence of 3*T(n)= 3*n(n-1)/2, where T(n) are the triangular numbers; n => 1.

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A309276 a(n) is the smallest divisor not yet in the sequence of 5*T(n)= 5*n(n-1)/2, where T(n) are the triangular numbers; n => 1.

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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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PARI

Python

Formula

A309387 a(n) is the smallest positive divisor not yet in the sequence of 7*A000217(n-1); n >= 1.

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Comments

Examples

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Crossrefs

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

A309202 Peter Munn showed in A111273 that if A111273(n)=m then if m is odd, n <= m, and if m is even, n <= 2m-1; a(n) is either m-n or 2m-1-n in the two cases.

A309275 a(n) is the smallest divisor not yet in the sequence of 3T(n)= 3n(n-1)/2, where T(n) are the triangular numbers; n => 1.

A309276 a(n) is the smallest divisor not yet in the sequence of 5T(n)= 5n(n-1)/2, where T(n) are the triangular numbers; n => 1.