A081371 -id:A081371 - cp's OEIS frontend

A076541 a(n) = Sum_{k=1..n} C(n,k) mod k.

0, 1, 2, 2, 3, 8, 9, 11, 6, 13, 14, 26, 27, 28, 32, 46, 47, 49, 50, 68, 81, 82, 83, 116, 102, 101, 105, 105, 106, 142, 143, 143, 153, 172, 165, 222, 223, 216, 259, 278, 279, 256, 257, 327, 394, 295, 296, 422, 374, 318, 328, 384, 385, 420, 469, 461, 465, 450, 451

Offset: 1

Benoit Cloitre, Oct 19 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Paolo P. Lava)
Amiram Eldar, Plot of a(n)/n^2 for n = 1..10^4.

Row sums of A081371.

a[n_] := Sum[Mod[Binomial[n, k], k], {k, 1, n}]; Array[a, 60] (* Amiram Eldar, Apr 26 2025 *)

PARI
```
a(n)=sum(k=1,n,binomial(n,k)%k)
```

a(n) seems to be asymptotic to c*n^2 with c around 0.18...

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).

A081372 a(n) is the number of j from 1 to n such that binomial(n,j) is divisible by j.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 3, 3, 1, 1, 3, 3, 5, 5, 4, 3, 1, 1, 2, 4, 2, 6, 7, 7, 2, 2, 8, 7, 6, 5, 6, 6, 4, 2, 4, 4, 2, 2, 2, 4, 2, 2, 4, 8, 12, 9, 8, 8, 12, 10, 12, 11, 8, 8, 6, 6, 3, 4, 14, 12, 9, 9, 9, 7, 4, 4, 7, 7, 6, 9, 11, 10, 4, 4, 6, 17, 15, 15, 11, 9, 6, 6, 9, 9, 6, 5, 6, 5, 2, 1, 4, 4, 9, 12

Offset: 1

Labos Elemer, Mar 21 2003

a(n) is the number of 0's in row n of A081371.

			n=10: C(10,j) mod j, j=1..10, is {0,1,0,2,2,0,1,5,1,1} includes 3 zeros, so a(10)=3.

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

Cf. A007318, A081371.

f:= n -> numboccur(0, map(j -> binomial(n,j) mod j, [$1..n])):
map(f, [$1..100]); # Robert Israel, Mar 19 2024

Flatten[Table[Count[Table[Mod[Binomial[n, j], j], {j, 1, n}], 0], {n, 1, 256}], 1]

Definition edited by Robert Israel, Mar 19 2024

A281908 a(n) = (Sum_{k=1..n} C(n,k) mod k) mod n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 2, 3, 6, 3, 3, 2, 1, 0, 2, 14, 13, 13, 12, 8, 18, 16, 14, 20, 2, 23, 24, 21, 19, 22, 19, 15, 21, 2, 25, 6, 1, 26, 25, 38, 33, 4, 42, 19, 34, 19, 14, 38, 31, 18, 22, 20, 14, 42, 29, 13, 9, 44, 38, 39, 31, 9, 4, 9, 25, 7, 63, 22, 2, 17, 7, 38, 27, 44, 25

Offset: 1

Michel Marcus, Feb 01 2017

nonn

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A076541, A081371, A280559.

a(n) = sum(k=1, n, binomial(n, k) % k) % n;

a(n) = A076541(n) mod n.

cp's OEIS Frontend

A076541 a(n) = Sum_{k=1..n} C(n,k) mod k.

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

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

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A081372 a(n) is the number of j from 1 to n such that binomial(n,j) is divisible by j.

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A281908 a(n) = (Sum_{k=1..n} C(n,k) mod k) mod n.

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