A081370 -id:A081370 - cp's OEIS frontend

A081369 Binomial(n^2, n) reduced mod n^2.

0, 2, 3, 12, 5, 12, 7, 56, 9, 40, 11, 108, 13, 84, 90, 240, 17, 144, 19, 80, 315, 220, 23, 72, 25, 312, 27, 560, 29, 0, 31, 992, 759, 544, 770, 720, 37, 684, 1053, 520, 41, 252, 43, 1408, 1125, 1012, 47, 1872, 49, 1200, 918, 624, 53, 1404, 2475, 2744, 2223, 1624, 59

Offset: 1

Labos Elemer, Mar 20 2003

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014062, A081370.

A081369:=n->binomial(n^2, n) mod n^2; seq(A081369(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[Mod[Binomial[n^2, n], n^2], {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n)=Mod[C(n^2, n), n^2]

A080216 a(n) is the largest value taken by binomial(n,j) mod j for j in [1..n].

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 4, 2, 5, 5, 7, 7, 7, 11, 8, 8, 9, 9, 13, 16, 11, 11, 15, 15, 13, 21, 18, 18, 18, 18, 18, 26, 26, 21, 25, 25, 21, 31, 28, 28, 29, 29, 31, 39, 27, 27, 36, 34, 31, 41, 34, 34, 45, 45, 36, 46, 46, 46, 43, 43, 41, 51, 40, 48, 52, 52, 52, 56, 44, 44, 52, 52, 57, 61

Offset: 1

Labos Elemer, Mar 21 2003

nonn

			n=13: {binomial(13,j) mod j, j=1..13} = {0,0,1,3,2,0,1,7,4,6,1,1,1}; maximum is 7, so a(13) = 7.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A007318, A080217, A081370, A081371.

Table[Max[Table[Mod[Binomial[n, j], j], {j, 1, n}]], {n, 1, 256}]

a(n) = vecmax(vector(n, j, binomial(n, j) % j)); \\ Michel Marcus, Jul 29 2017

def a(n):
    return max([binomial(n,j)%j for j in range(1, n+1)])  # Robin Visser, Nov 26 2023

a(n) = max_{j=1..n} binomial(n,j) mod j.

A080217 a(n) is the number of distinct values taken by binomial(n,j) mod j for j in [1..n].

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 4, 3, 4, 4, 7, 7, 6, 5, 7, 7, 7, 7, 9, 11, 12, 12, 12, 11, 11, 10, 11, 11, 12, 12, 12, 11, 12, 12, 13, 13, 13, 17, 18, 18, 15, 15, 18, 21, 17, 17, 19, 19, 18, 17, 16, 16, 20, 20, 23, 25, 23, 23, 26, 26, 24, 22, 24, 24, 27, 27, 27, 25, 28, 28, 30, 30, 32, 31, 30

Offset: 1

Labos Elemer, Mar 21 2003

nonn

			n=14: {binomial(14,j) mod j, j=1..14} = {0,1,1,1,2,3,2,3,4,1,1,7,1,1} includes six distinct residues (0,1,2,3,4,7) so a(14) = 6.

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from Vincenzo Librandi).

Cf. A007318, A080216, A081370, A081371.

Table[Length[Union[Table[Mod[Binomial[n, j], j], {j, 1, n}]]], {n, 1, 256}]

a(n) = #vecsort(vector(n, j, binomial(n, j) % j), ,8); \\ Michel Marcus, Jul 29 2017

def a(n):
    return len(set([binomial(n,j)%j for j in range(1, n+1)]))  # Robin Visser, Nov 26 2023

a(n) = Card(Union{j=1..n} binomial(n,j) mod j).

A371472 Least positive number k such that binomial(k^2,k) is divisible by n^2.

Original entry on oeis.org

1, 3, 8, 6, 9, 8, 10, 23, 16, 16, 16, 14, 23, 10, 9, 23, 45, 16, 27, 16, 34, 16, 33, 23, 94, 23, 105, 20, 77, 16, 54, 91, 16, 45, 19, 16, 83, 27, 23, 30, 58, 34, 114, 16, 16, 40, 133, 23, 130, 94, 45, 23, 75, 105, 16, 38, 27, 77, 145, 16, 106, 54, 47, 91, 49, 16, 190, 45, 80, 19, 123, 47, 283, 83, 94, 27, 40, 23, 137

Offset: 1

Seiichi Manyama, Mar 24 2024

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014062, A081370, A371306.

a(n) = my(k=1); while(binomial(k^2, k)%n^2>0, k++); k;

A371474 Numbers k such that binomial(k^2,k) == 0 (mod k^3).

Original entry on oeis.org

1, 2184, 6552, 12870, 13860, 19530, 23100, 33660, 40755, 47880, 51051, 58995, 81396, 88920, 101010, 113553, 114114, 114855, 121800, 125970, 136136, 141372, 142290, 142324, 145860, 150535, 154583, 157080, 158928, 164424, 171080, 180180, 193732, 195104, 197340, 214890, 225680, 229908, 230230, 230724, 238602, 243542, 249964, 253080, 257712, 267960, 284867, 291720, 294525, 297414, 300696

Offset: 1

Seiichi Manyama, Mar 24 2024

nonn

Cf. A014062, A081370.

PARI
```
isok(n) = binomial(n^2, n)%n^3==0;
```

from itertools import count, islice
from math import comb
def A371474_gen(): # generator of terms
    return filter(lambda k:not comb(k**2,k)%(k**3),count(1))
A371474_list = list(islice(A371474_gen(),3)) # Chai Wah Wu, Mar 25 2024

More terms from Vaclav Kotesovec, Mar 26 2024

cp's OEIS Frontend

A081369 Binomial(n^2, n) reduced mod n^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A080216 a(n) is the largest value taken by binomial(n,j) mod j for j in [1..n].

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A080217 a(n) is the number of distinct values taken by binomial(n,j) mod j for j in [1..n].

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A371472 Least positive number k such that binomial(k^2,k) is divisible by n^2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A371474 Numbers k such that binomial(k^2,k) == 0 (mod k^3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Extensions