A133885 -id:A133885 - cp's OEIS frontend

A133625 Binomial(n+p, n) mod n where p=5.

0, 1, 2, 2, 2, 0, 1, 7, 4, 3, 1, 8, 1, 8, 9, 13, 1, 7, 1, 10, 8, 12, 1, 3, 6, 1, 10, 8, 1, 2, 1, 25, 12, 1, 8, 22, 1, 20, 14, 39, 1, 15, 1, 12, 25, 24, 1, 5, 1, 11, 18, 14, 1, 46, 12, 43, 20, 1, 1, 48, 1, 32, 22, 49, 14, 23, 1, 18, 24, 50, 1, 7, 1, 1, 41, 20, 1, 66, 1, 77, 28, 1, 1, 50, 18, 44

Offset: 1

Hieronymus Fischer, Sep 30 2007

Let d(m)...d(2)d(1)d(0) be the base-n representation of n+p. The relation a(n)=d(1) holds, if n is a prime index. For this reason there are infinitely many terms which are equal to 1.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 240.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362688, A362689.

Table[Mod[Binomial[n+5,n],n],{n,90}] (* Harvey P. Dale, Oct 02 2015 *)

a(n)=binomial(n+5,5) mod n.
a(n)=1 if n is a prime > 5, since binomial(n+5,n)==(1+floor(5/n))(mod n), provided n is a prime.
From Chai Wah Wu, May 26 2016: (Start)
a(n) = (n^5 + 15*n^4 + 85*n^3 + 105*n^2 + 34*n + 120)/120 mod n.
For n > 6:
if n mod 120 == 0, then a(n) = 17*n/60 + 1.
if n mod 120 is in {1, 2, 7, 11, 13, 17, 19, 23, 26, 29, 31, 34, 37, 41, 43, 47, 49, 53, 58, 59, 61, 67, 71, 73, 74, 77, 79, 82, 83, 89, 91, 97, 98, 101, 103, 106, 107, 109, 113, 119}, then a(n) = 1.
if n mod 120 is in {3, 9, 18, 21, 27, 33, 39, 42, 51, 57, 63, 66, 69, 81, 87, 93, 99, 111, 114, 117}, then a(n) = n/3 + 1.
if n mod 120 is in {4, 28, 44, 52, 68, 76, 92, 116}, then a(n) = n/4 + 1.
if n mod 120 is in {5, 10, 25, 35, 50, 55, 65, 85, 95, 115}, then a(n) = n/5 + 1.
if n mod 120 is in {6, 54, 78, 102}, then a(n) = 5*n/6 + 1.
if n mod 120 is in {8, 16, 32, 56, 64, 88, 104, 112}, then a(n) = 3*n/4 + 1.
if n mod 120 is in {12, 36, 84, 108}, then a(n) = 7*n/12 + 1.
if n mod 120 is in {14, 22, 38, 46, 62, 86, 94, 118}, then a(n) = n/2 + 1.
if n mod 120 is in {15, 45, 75, 90, 105}, then a(n) = 8*n/15 + 1.
if n mod 120 is in {20, 100}, then a(n) = 9*n/20 + 1.
if n mod 120 is in {24, 48, 72, 96}, then a(n) = n/12 + 1.
if n mod 120 == 30, then a(n) = n/30 + 1.
if n mod 120 is in {40, 80}, then a(n) = 19*n/20 + 1.
if n mod 120 == 60, then a(n) = 47*n/60 + 1.
if n mod 120 is in {70, 110}, then a(n) = 7*n/10 + 1.
(End)
For n > 246, a(n) = 2*a(n-120) - a(n-240). - Ray Chandler, Apr 23 2023

A133635 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=5.

Original entry on oeis.org

26, 34, 49, 58, 74, 77, 82, 91, 98, 106, 119, 121, 122, 133, 143, 146, 154, 161, 169, 178, 187, 194, 202, 203, 209, 217, 218, 221, 226, 242, 247, 253, 259, 266, 274, 287, 289, 298, 299, 301, 314, 319, 322, 323, 329, 338, 341, 343, 346, 361, 362, 371, 377, 386

Offset: 1

Hieronymus Fischer, Sep 30 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910.

nn=400;With[{c=Complement[Range[nn],Prime[Range[PrimePi[nn]]]]}, Select[ c,Mod[Binomial[#+5,#],#]==1&]] (* Harvey P. Dale, Sep 24 2012 *)

from itertools import count, islice
from math import comb
from sympy import isprime
def A133635_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:comb(k+5,k)%k==1 and not isprime(k),count(max(startvalue,1)))
A133635_list = list(islice(A133635_gen(),30)) # Chai Wah Wu, Feb 22 2023

A133875 n modulo 5 repeated 5 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 5^2 = 25.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1).

Cf. A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133885, A133880, A133890, A133900, A133910.

[(1 + Floor(n/5)) mod 5 : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

A133875:=n->((1+floor(n/5)) mod 5); seq(A133875(n), n=0..100); # Wesley Ivan Hurt, Jun 06 2014

Table[Mod[1 + Floor[n/5], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 06 2014 *)
LinearRecurrence[{1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1},{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0},120] (* Harvey P. Dale, Dec 14 2017 *)

a(n) = (1 + floor(n/5)) mod 5.
a(n) = A010874(A002266(n+5)).
a(n) = 1 + floor(n/5) - 5*floor((n+5)/25).
a(n) = (((n+5) mod 25) - (n mod 5)) / 5.
a(n) = ((n + 5 - (n mod 5)) / 5) mod 5.
a(n) = A010874((n + 5 - A010874(n))/5).
a(n) = binomial(n+5, n) mod 5 = binomial(n+5, 5) mod 5.
a(n) = +a(n-1) -a(n-5) +a(n-6) -a(n-10) +a(n-11) -a(n-15) +a(n-16) -a(n-20) +a(n-21). - R. J. Mathar, Sep 03 2011
G.f.: ( 1+2*x^5+3*x^10+4*x^15 ) / ( (1-x)*(x^20+x^15+x^10+x^5+1) ). - R. J. Mathar, Sep 03 2011

A362686 Binomial(n+p, n) mod n where p=6.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 1, 3, 1, 8, 1, 0, 1, 8, 9, 5, 1, 10, 1, 10, 15, 12, 1, 15, 6, 14, 1, 8, 1, 12, 1, 9, 12, 18, 8, 10, 1, 20, 27, 19, 1, 36, 1, 12, 10, 24, 1, 45, 1, 36, 18, 14, 1, 28, 12, 15, 39, 30, 1, 48, 1, 32, 1, 17, 14, 12, 1, 18, 24, 50, 1, 19, 1, 38

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 1440.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362687, A362688, A362689.

Table[Mod[Binomial[n+6, n], n], {n, 90}]

a(n)=binomial(n+6,n) mod n.

For n > 1452, a(n) = 2*a(n-720) - a(n-1440).

A362687 Binomial(n+p, n) mod n where p=7.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 2, 3, 1, 8, 1, 0, 1, 10, 9, 5, 1, 10, 1, 10, 18, 12, 1, 15, 6, 14, 1, 12, 1, 12, 1, 9, 12, 18, 13, 10, 1, 20, 27, 19, 1, 0, 1, 12, 10, 24, 1, 45, 8, 36, 18, 14, 1, 28, 12, 23, 39, 30, 1, 48, 1, 32, 10, 17, 14, 12, 1, 18, 24, 60, 1, 19, 1

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 10080.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362688, A362689.

Mathematica
```
Table[Mod[Binomial[n+7,n],n],{n,90}]
```

a(n)=binomial(n+7,n) mod n.

For n > 10122, a(n) = 2*a(n-5040) - a(n-10080).

A362688 Binomial(n+p, n) mod n where p=8.

Original entry on oeis.org

0, 1, 0, 3, 2, 3, 2, 6, 1, 8, 1, 6, 1, 10, 9, 15, 1, 1, 1, 5, 18, 1, 1, 12, 6, 14, 1, 12, 1, 12, 1, 13, 12, 1, 13, 19, 1, 1, 27, 34, 1, 0, 1, 34, 10, 24, 1, 27, 8, 11, 18, 1, 1, 1, 12, 16, 39, 30, 1, 48, 1, 32, 10, 25, 14, 45, 1, 35, 24, 25, 1, 46, 1, 38, 66

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 645120.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362689.

Mathematica
```
Table[Mod[Binomial[n+8,n],n],{n,90}]
```

a(n)=binomial(n+8,n) mod n.

For n > 645240, a(n) = 2*a(n-322560) - a(n-645120).

A362689 Binomial(n+p, n) mod n where p=9.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 2, 6, 2, 8, 1, 2, 1, 10, 14, 15, 1, 3, 1, 5, 18, 1, 1, 12, 6, 14, 4, 12, 1, 22, 1, 13, 1, 1, 13, 23, 1, 1, 14, 34, 1, 14, 1, 34, 15, 24, 1, 27, 8, 11, 18, 1, 1, 7, 12, 16, 1, 30, 1, 28, 1, 32, 17, 25, 14, 23, 1, 35, 47, 25, 1, 54, 1, 38, 66

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 5806080.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362688.

Mathematica
```
Table[Mod[Binomial[n+9,n],n],{n,90}]
```

a(n)=binomial(n+9,n) mod n.

For n > 5806081, a(n) = 2*a(n-2903040) - a(n-5806080).

A133895 Numbers m such that binomial(m+5,m) mod 5 = 0.

Original entry on oeis.org

20, 21, 22, 23, 24, 45, 46, 47, 48, 49, 70, 71, 72, 73, 74, 95, 96, 97, 98, 99, 120, 121, 122, 123, 124, 145, 146, 147, 148, 149, 170, 171, 172, 173, 174, 195, 196, 197, 198, 199, 220, 221, 222, 223, 224, 245, 246, 247, 248, 249, 270, 271, 272, 273, 274, 295

Offset: 0

Hieronymus Fischer, Oct 20 2007

Also numbers m such that floor(1+(m/5)) mod 5 = 0.

Partial sums of the sequence 20,1,1,1,1,21,1,1,1,1, 21, ... which has period 5.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A000040, A133620, A133621, A133623, A133630, A133635.

Cf. A133875, A133885, A133890, A133900, A133910.

a(n)=5n+20-4*(n mod 5).
G.f.: g(x)=(20+x+x^2+x^3+x^4+x^5)/((1-x^5)(1-x)).
G.f.: g(x)=(20-19x-x^6) /((1-x^5)(1-x)^2).

cp's OEIS Frontend

A133625 Binomial(n+p, n) mod n where p=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A133635 Nonprime numbers k such that binomial(k+p,k) mod k = 1, where p=5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A133875 n modulo 5 repeated 5 times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A362686 Binomial(n+p, n) mod n where p=6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A362687 Binomial(n+p, n) mod n where p=7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A362688 Binomial(n+p, n) mod n where p=8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A362689 Binomial(n+p, n) mod n where p=9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A133895 Numbers m such that binomial(m+5,m) mod 5 = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula