A133625 -id:A133625 - cp's OEIS frontend

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

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

A133623 Binomial(n+p, n) mod n where p=3.

Original entry on oeis.org

0, 0, 2, 3, 1, 0, 1, 5, 4, 6, 1, 11, 1, 8, 6, 9, 1, 16, 1, 11, 8, 12, 1, 21, 1, 14, 10, 15, 1, 26, 1, 17, 12, 18, 1, 31, 1, 20, 14, 21, 1, 36, 1, 23, 16, 24, 1, 41, 1, 26, 18, 27, 1, 46, 1, 29, 20, 30, 1, 51, 1, 32, 22, 33, 1, 56, 1, 35, 24, 36, 1, 61, 1, 38, 26, 39, 1, 66, 1, 41, 28, 42, 1

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.

Appears to satisfy the recurrence: a(n) = -2*a(n-1) - a(n-2) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 2*a(n-7) - a(n-8) for n > 14. - Chai Wah Wu, May 25 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-1,2,4,2,-1,-2,-1).

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

Cf. A133873, A133883, A133880, A133890, A133900, A133910.

Table[Mod[Binomial[n+3,n],n],{n,90}] (* Harvey P. Dale, Nov 22 2011 *)

a(n) = binomial(n+3,3) mod n.
a(n)=1 if n is a prime > 3, since binomial(n+3,n)==(1+floor(3/n))(mod n), provided n is a prime.
From Chai Wah Wu, May 26 2016: (Start)
a(n) = (n^3 + 5*n + 6)/6 mod n.
For n > 6:
if n mod 6 == 0, then a(n) = 5*n/6 + 1.
if n mod 6 is in {1, 5}, then a(n) = 1.
if n mod 6 is in {2, 4}, then a(n) = n/2 + 1.
if n mod 6 == 3, then a(n) = n/3 + 1.
(End)

A133622 a(n) = 1 if n is odd, a(n) = n/2+1 if n is even.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44, 1

Offset: 1

Hieronymus Fischer, Sep 30 2007

a(n) is the count of terms a(n+1) present so far in the sequence, with a(n+1) included in the count; example: a(1) = 1 "says" that there is 1 term "2" so far in the sequence; a(2) = 2 "says" that there are 2 terms "1" so far in the sequence... etc. This comment was inspired by A039617. - Eric Angelini, Mar 03 2020

Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -1).

Cf. A133620, A133621, A133623, A133624, A133625, A133630, A038509, A133634-A133636, A133872, A133882, A133880, A133890, A133900, A133910, A165157 (partial sums).

Other related sequences: A000217, A007879, A057979.

import Data.List (transpose)
a133622 n = (1 - m) * n' + 1 where (n', m) = divMod n 2
a133622_list = concat $ transpose [[1, 1 ..], [2 ..]]
-- Reinhard Zumkeller, Feb 20 2015

seq([1,n][],n=2..100); # Robert Israel, May 27 2016

Riffle[Range[2,50],1,{1,-1,2}] (* Harvey P. Dale, Jan 19 2013 *)

a(n)=if(n%2,1,n/2+1) \\ Charles R Greathouse IV, Sep 02 2015

a(n)=1+(binomial(n+1,2)mod n)=1+(binomial(n+1,n-1)mod n).
a(n)=binomial(n+2,2) mod n = binomial(n+2,n) mod n for n>2.
a(n)=1+(1+(-1)^n)*n/4.
a(n)=1+(A000217(n) mod n).
a(n)=a(n-2)+1, if n is even, a(n)=a(n-2) if n is odd.
a(n)=a(n-2)+1-(n mod 2)=a(n-2)+(1+(-1)^n)/2 for n>2.
a(n)=(a(n-3)+a(n-2))/a(n-1) for n>3.
G.f.: g(x)=x(1+2x-x^2-x^3)/(1-x^2)^2.
G.f.: (Q(0)-1-x)/x^2, where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013
a(n) = 2*a(n-2)-a(n-4) for n > 4. - Chai Wah Wu, May 26 2016
E.g.f.: exp(x) - 1 + x*sinh(x)/2. - Robert Israel, May 27 2016

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

A133885 Binomial(n+5,n) mod 5^2.

Original entry on oeis.org

1, 6, 21, 6, 1, 2, 12, 17, 12, 2, 3, 18, 13, 18, 3, 4, 24, 9, 24, 4, 5, 5, 5, 5, 5, 6, 11, 1, 11, 6, 7, 17, 22, 17, 7, 8, 23, 18, 23, 8, 9, 4, 14, 4, 9, 10, 10, 10, 10, 10, 11, 16, 6, 16, 11, 12, 22, 2, 22, 12, 13, 3, 23, 3, 13, 14, 9, 19, 9, 14, 15, 15, 15, 15, 15, 16, 21, 11, 21, 16, 17

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 5^3=125.

Ray Chandler, Table of n, a(n) for n = 0..1000
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, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133880, A133890, A133900, A133910.
For the sequence regarding binomial(n+5, n) mod 5 see A133875.

Table[Mod[Binomial[n+5,n],25],{n,0,90}] (* Harvey P. Dale, Jan 12 2023 *)

a(n)=binomial(n+5,5) mod 5^2.

G.f. g(x)=sum{0<=k<125, a(k)*x^k}/(1-x^125).

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).

A133882 a(n) = binomial(n+2,n) mod 2^2.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 2^3 = 8.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1, 1, -1, 1).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133872, A133880, A133890, A133900, A133910.
For the sequence regarding "binomial(n+2, n) mod 2" see A133872.
A105198 shifted once left.

Table[Mod[Binomial[n+2,n],4],{n,0,120}]  (* Harvey P. Dale, Apr 16 2011 *)

A133882(n) = (binomial(n+2,n) % 4); \\ Antti Karttunen, Aug 10 2017

a(n) = binomial(n+2,2) mod 2^2.
G.f.: (1 + 3*x + 2*x^2 + 2*x^3 + 3*x^4 + x^5)/(1-x^8).
G.f.: (1+x)*(1+2*x+2*x^3+x^4)/(1-x^8) = (1+2*x+2*x^3+x^4)/((1-x)*(1+x^2)*(1+x^4)).
a(n) = A105198(n+1). - R. J. Mathar, Jun 08 2008

cp's OEIS Frontend

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

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A133623 Binomial(n+p, n) mod n where p=3.

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A133622 a(n) = 1 if n is odd, a(n) = n/2+1 if n is even.

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A133875 n modulo 5 repeated 5 times.

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A133885 Binomial(n+5,n) mod 5^2.

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

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

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

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