A038509 -id:A038509 - cp's OEIS frontend

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

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

A133883 a(n) = binomial(n+3,n) mod 3^2.

Original entry on oeis.org

1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 3^3 = 27.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133873, A133880, A133890, A133900, A133910.
For the sequence regarding "Binomial(n+3, n) mod 3" see A133873.

[Binomial(n+3,n) mod 9: n in [0..60]]; // Vincenzo Librandi, Jul 20 2016

Table[Mod[Binomial[n + 3, n], 9], {n, 0, 120}] (* or *)
CoefficientList[Series[(1 + 3 x - 3 x^2 + 2 x^3 + 9 x^4 - 9 x^5 + 3 x^6 + 9 x^7 - 9 x^8 + 4 x^9 + 12 x^10 - 12 x^11 + 5 x^12 + 9 x^13 - 9 x^14 + 6 x^15 + 9 x^16 - 9 x^17 + 7 x^18 + 3 x^19 - 3 x^20 + 8 x^21)/((1 - x) (1 + x^3 + x^6) (1 + x^9 + x^18)), {x, 0, 120}], x] (* Michael De Vlieger, Jul 19 2016 *)

Vec((1 +3*x -3*x^2 +2*x^3 +9*x^4 -9*x^5 +3*x^6 +9*x^7 -9*x^8 +4*x^9 +12*x^10 -12*x^11 +5*x^12 +9*x^13 -9*x^14 +6*x^15 +9*x^16 -9*x^17 +7*x^18 +3*x^19 -3*x^20 +8*x^21) / ((1 -x)*(1 +x^3 +x^6)*(1 +x^9 +x^18)) + O(x^200)) \\ Colin Barker, Jul 19 2016

a(n) = binomial(n+3,3) mod 3^2.

G.f.: (1 +3*x -3*x^2 +2*x^3 +9*x^4 -9*x^5 +3*x^6 +9*x^7 -9*x^8 +4*x^9 +12*x^10 -12*x^11 +5*x^12 +9*x^13 -9*x^14 +6*x^15 +9*x^16 -9*x^17 +7*x^18 +3*x^19 -3*x^20 +8*x^21) / ((1 -x)*(1 +x^3 +x^6)*(1 +x^9 +x^18)). - Colin Barker, Jul 19 2016

Corrected g.f. by Colin Barker, Jul 19 2016

A133886 a(n) = binomial(n+6,n) mod 6.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 2*6^2 = 72.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1).

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

Cf. A133876, A133880, A133890, A133900, A133910.

A133886:=n->binomial(n+6,6) mod 6; seq(A133886(n), n=0..100); # Wesley Ivan Hurt, Apr 30 2014

Table[Mod[Binomial[n+6,n],6],{n,0,110}] (* Harvey P. Dale, May 04 2013 *)

a(n) = binomial(n+6,6) mod 6.
G.f.: g(x) = (1+x+4*x^2-6*x^9-6*x^56+4*x^63+x^64+x^65+3*x^8*(1+x)(1-x^56)/(1-x^8)+4*x^9(1+x+x^2)(1-x^54)/(1-x^9))/(1-x^72).
a(n) = a(n-1)-a(n-2)+a(n-8)+a(n-11)-a(n-17)-a(n-20)-a(n-24)+a(n-25)+a(n-29)+ a(n-32)- a(n-38)-a(n-41)+a(n-47)-a(n-48)+a(n-49). - Harvey P. Dale, May 04 2013

A067793 Nonprimes n such that phi(n) > 2n/3.

Original entry on oeis.org

1, 25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 391, 395, 403, 407, 413, 415, 425, 427

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Differs from A038509 in the first entry and in addition as documented in A069043. - R. J. Mathar, Sep 30 2008
It appears that a(n) lists the composite values of n which satisfy the condition sum(k^2,k=1..n) mod n = A000330(n) mod n = A215573(n) = 0. - Gary Detlefs, Nov 16 2011
Conjecture: Odd composite n such that (n^2 + 8) mod 3 = 0. (All primes > 3 meet this criterion). - Gary Detlefs, May 03 2012
Both conjectures are wrong.  The first counterexample is 385. - Robert Israel, May 17 2017
The semiprime numbers p * q, p, q > 3, are terms. - Marius A. Burtea, Oct 01 2019

			10 is not in the list because phi(10) = 4 < 2*10/3. 25 is in the list because phi(25) = 20 > 2*25/3.

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

Cf. A166362.

[k:k in [1..400]| not IsPrime(k) and EulerPhi(k) gt 2*k/3]; // Marius A. Burtea, Oct 01 2019

select(n -> not isprime(n) and numtheory:-phi(n) > 2*n/3, [$1..1000]); # Robert Israel, May 17 2017

Select[Range[1000], ! PrimeQ[#] && EulerPhi[#] > 2 #/3 &] (* T. D. Noe, Nov 02 2011 *)

lista(nn) = {for (n=1, nn, if (!isprime(n) && (eulerphi(n)/n > 2/3), print1(n, ", ")););} \\ Michel Marcus, Jul 05 2015

Definition clarified by Michel Marcus, Jul 05 2015

Incorrect Maple program removed by Robert Israel, May 17 2017

A133873 n modulo 3 repeated 3 times.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 3^2=9.

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

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

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

G.f.: (1 + 2*x^3)*(1 - x^3)/((1 - x)*(1 - x^9)).
a(n) = (1 + floor(n/3)) mod 3.
a(n) = A010872(A002264(n+3)).
a(n) = 1+floor(n/3)-3*floor((n+3)/9).
a(n) = (((n+3) mod 9)-(n mod 3))/3.
a(n) = ((n+3-(n mod 3))/3) mod 3.
a(n) = binomial(n+3,n) mod 3 = binomial(n+3,3) mod 3.

A133874 n modulo 4 repeated 4 times.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

nonn

Periodic with length 4^2 = 16.

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

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

Cf. A133884, A133880, A133890, A133900, A133910.

Flatten[Table[Table[Mod[n,4],{4}],{n,30}]] (* Harvey P. Dale, Dec 22 2013 *)

def A133874(n): return 1+(n>>2)&3 # Chai Wah Wu, Jan 18 2023

a(n) = (1 + floor(n/4)) mod 4.
a(n) = A010873(A002265(n+4)).
a(n) = 1 + floor(n/4) - 4*floor((n+4)/16).
a(n) = (((n+4) mod 16) - (n mod 4))/4.
a(n) = ((n + 4 - (n mod 4))/4) mod 4.
G.f. g(x) = (1 + x + x^2 + x^3 + 2x^4 + 2x^5 + 2x^6 + 2x^7 + 3x^8 + 3x^9 + 3x^10 + 3x^11)/(1-x^16).
G.f. g(x) = ((1-x^4)*(1+2x^4+3x^8))/((1-x)*(1-x^16)).
G.f. g(x) = (3x^16-4x^12+1)/((1-x)*(1-x^4)*(1-x^16)).
G.f. g(x) = (1+2x^4+3x^8)/((1-x)*(1+x^4)*(1+x^8)).

A133884 a(n) = binomial(n+4,n) mod 4.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 4^2=16.

			For n=2, binomial(6,2) = 6*5/2 = 15, which is 3 (mod 4) so a(2) = 3. - _Michael B. Porter_, Jul 19 2016

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

Cf. A000040, A133630, A038509.
Cf. A133620, A133621, A133622, A133623, A133624, A133625
Cf. A133634, A133635, A133636.
Cf. A133874, A133880, A133890, A133900, A133910.

[Binomial(n+4,n) mod 4: n in [0..100]]; // Vincenzo Librandi, Jul 15 2016

Table[Mod[Binomial[n + 4, 4], 4], {n, 0, 100}] (* Vincenzo Librandi, Jul 15 2016 *)

a(n) = binomial(n+4,4) mod 4.
G.f.: (1 + x + 3*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + x^10 + x^11)/(1 - x^16) = (1 + 2*x^2 + 2*x^6 + x^8)/((1 - x)*(1 + x^4)*(1 + x^8)).
a(n) = A000505(n+5) mod 4. - John M. Campbell, Jul 14 2016
a(n) = A000506(n+6) mod 4. - John M. Campbell, Jul 15 2016

G.f. corrected by Bruno Berselli, Jul 19 2016

cp's OEIS Frontend

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

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

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A133883 a(n) = binomial(n+3,n) mod 3^2.

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A133886 a(n) = binomial(n+6,n) mod 6.

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A067793 Nonprimes n such that phi(n) > 2n/3.

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A133873 n modulo 3 repeated 3 times.

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