A133883 -id:A133883 - cp's OEIS frontend

A133900 a(n) = period of the sequence {b(m), m>=0}, defined by b(m):=binomial(m+n,n) mod n.

1, 4, 9, 16, 25, 72, 49, 64, 81, 400, 121, 864, 169, 784, 675, 256, 289, 2592, 361, 1600, 1323, 3872, 529, 3456, 625, 5408, 729, 3136, 841, 324000, 961, 1024, 9801, 18496, 6125, 31104, 1369, 23104, 13689, 32000, 1681, 254016, 1849, 15488, 30375, 33856

Offset: 1

Hieronymus Fischer, Oct 15 2007, Oct 20 2007

nonn

This is the analog of the sequence of Pisano periods (A001175) for binomial factors.
n^2 always divides a(n).
A prime p is a factor of a(n) if and only if it is a factor of n (i.e., a(n) and n have the same prime factors).

			a(3)=9 since binomial(m+3,3) mod 3, m>=0, is periodic with period length 3^2=9 (see A133883).
a(6)=72 since binomial(m+6,6) mod 6, m>=0, is periodic with period length 4*6^2=72 (see A133886).

Hieronymus Fischer, Table of n, a(n) for n = 1..111

Cf. A000040, A001175, A133872-A133880, A133882-A133890, A133910.
Cf. A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133905.

a(n)=n^2 if n is a prime or a power of a prime.

A038509 Composite numbers congruent to +-1 mod 6.

Original entry on oeis.org

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, 385

Offset: 1

Jeff Burch

Or, composite numbers with smallest prime factor >= 5.
Or, nonprime numbers n such that binomial(n+3, 3) mod n == 1. - Hieronymus Fischer, Sep 30 2007
Note that the primes > 3 are congruent to +-1 mod 6.
This sequence differs from A067793 (composite n such that phi(n) > 2n/3) starting at 385. Numbers in this sequence but not in A067793 are 385, 455, 595, 665, 805, 1015, 1085, 1925, 2275, 2695, etc. See A069043. - R. J. Mathar, Jun 08 2008 and Zak Seidov, Nov 02 2011
Intersection of A002808 and A007310. - Reinhard Zumkeller, Jun 30 2012
The product (24/25) * (36/35) * (48/49) * (54/55) * (66/65) * (78/77) * (84/85) * (90/91) * ... * ((6*k)/a(n)) * ... = Pi^2/(6*sqrt(3)), where 6*k is the nearest number to a(n), with k in A067611 but not in A002822. (See A258414.) - Dimitris Valianatos, Mar 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A171993 (nonprimes of the form 3*k+-1).
Cf. A000040, A133620-A133625, A133630, A133634-A133636.
Cf. A133873, A133883, A133880, A133890, A133900, A133910.
Cf. A069043, A067793 (composite n such that phi(n) > 2n/3).

a038509 n = a038509_list !! (n-1)
a038509_list = [x | x <- a002808_list, gcd x 6 == 1]
-- Reinhard Zumkeller, Aug 05 2014, Jun 30 2012

A038509 := proc(n)
    option remember;
    if n = 1 then
        25;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,6) in {1,5} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A038509(n),n=1..30) ; # R. J. Mathar, Feb 28 2020

Select[Range[1000], FactorInteger[#][[1,1]] >= 5 && ! PrimeQ[#] &] (* Robert G. Wilson v, Dec 19 2009 *)
With[{nn=400},Select[Rest[Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]], MemberQ[ {1,5},Mod[#,6]]&]] (* Harvey P. Dale, Feb 21 2013 *)
Select[Range[400],CompositeQ[#]&&MemberQ[{1,5},Mod[#,6]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

is(n)=gcd(n,6)==1 && !isprime(n) && n>7 \\ Charles R Greathouse IV, Nov 20 2012

a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012

More terms from Robert G. Wilson v, Dec 19 2009

Entry revised by N. J. A. Sloane, Dec 31 2011, at the suggestion of Gary Detlefs

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)

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.

A133893 Numbers m such that binomial(m+3,m) mod 3 = 0.

Original entry on oeis.org

6, 7, 8, 15, 16, 17, 24, 25, 26, 33, 34, 35, 42, 43, 44, 51, 52, 53, 60, 61, 62, 69, 70, 71, 78, 79, 80, 87, 88, 89, 96, 97, 98, 105, 106, 107, 114, 115, 116, 123, 124, 125, 132, 133, 134, 141, 142, 143, 150, 151, 152, 159, 160, 161, 168, 169, 170, 177, 178, 179, 186

Offset: 0

Hieronymus Fischer, Oct 20 2007

nonn

Also numbers m such that floor(1+(m/3)) mod 3 = 0.

Partial sums of the sequence 6,1,1,7,1,1,7,1,1,7, ... which has period 3.

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

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

Select[Range[200],Mod[Binomial[#+3,#],3]==0&] (* Harvey P. Dale, Aug 27 2023 *)

a(n)=3n+6-2*(n mod 3).
G.f.: g(x)=6/(1-x)+x(1+x+7x^2)/((1-x^3)(1-x)) = (6+x+x^2+x^3)/((1-x^3)(1-x)).
G.f.: g(x)=(6-5x-x^4)/((1-x^3)(1-x)^2).

cp's OEIS Frontend

A133900 a(n) = period of the sequence {b(m), m>=0}, defined by b(m):=binomial(m+n,n) mod n.

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A038509 Composite numbers congruent to +-1 mod 6.

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

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A133873 n modulo 3 repeated 3 times.

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A133893 Numbers m such that binomial(m+3,m) mod 3 = 0.

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