A109004 -id:A109004 - cp's OEIS frontend

A109013 a(n) = gcd(n,10).

10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1

Offset: 0

Mitch Harris

Cf. A109004.

GCD[Range[0,100],10] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = 1 + [2|n] + 4*[5|n] + 4*[10|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-10).
Multiplicative with a(p^e, 10) = gcd(p^e, 10). - David W. Wilson, Jun 12 2005
G.f.: ( -10 - x - 2*x^2 - x^3 - 2*x^4 - 5*x^5 - 2*x^6 - x^7 - 2*x^8 - x^9 ) / ( (x-1)*(1+x)*(x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x+1) ). - R. J. Mathar, Apr 04 2011
Dirichlet g.f.: zeta(s)*(1 + 1/2^s + 4/5^s + 4/10^s). - R. J. Mathar, Apr 04 2011
a(n) = ((n-1) mod 2 + 1)*(4*floor(((n-1) mod 5)/4) + 1). - Gary Detlefs, Dec 28 2011

A109010 a(n) = gcd(n,7).

Original entry on oeis.org

7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1

Offset: 0

Mitch Harris

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

Cf. A109004.

A109010:=n->gcd(n,7): seq(A109010(n), n=0..150); # Wesley Ivan Hurt, Apr 27 2017

GCD[Range[0,100],7] (* or *) PadRight[{},120,{7,1,1,1,1,1,1}] (* Harvey P. Dale, Apr 26 2018 *)

a(n) = 1 + 6*[7|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-7).
Multiplicative with a(p^e, 7) = gcd(p^e, 7). - David W. Wilson, Jun 12 2005
From R. J. Mathar, Apr 04 2011: (Start)
Dirichlet g.f.: zeta(s)*(1 + 6/7^s).
G.f.: (-7 - x - x^2 - x^3 - x^4 - x^5 - x^6) / ((x-1)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). (End)
a(n) = 6*floor(((n-1) mod 7)/6) + 1. - Gary Detlefs, Dec 28 2011

A109011 a(n) = gcd(n,8).

Original entry on oeis.org

8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4

Offset: 0

Mitch Harris

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

Cf. A010877, A109004.

a[n_]:= GCD[n,8]; Array[a, 100, 0] (* Stefano Spezia, Nov 19 2018 *)

a(n) = gcd(n, 8) \\ David A. Corneth, Nov 19 2018

a(n) = 1 + [2|n] + 2*[4|n] + 4*[8|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-8).
Multiplicative with a(p^e) = gcd(p^e, 8). - David W. Wilson, Jun 12 2005
G.f.: ( -8 - x - 2*x^2 - x^3 - 4*x^4 - x^5 - 2*x^6 - x^7 ) / ( (x-1)*(1+x)*(x^2+1)*(x^4+1) ). - R. J. Mathar, Apr 04 2011
Dirichlet g.f.: zeta(s)*(1 + 1/2^s + 2/4^s + 4/8^s). - R. J. Mathar, Apr 04 2011
a(n) = 2^(-(101*m^7 - 2464*m^6 + 23786*m^ 5 -115360*m^4 + 293909*m^3 - 371056*m^2 + 186204*m - 15120)/5040) where m = (n mod 8). - Luce ETIENNE, Nov 18 2018

A109014 a(n) = gcd(n,11).

Original entry on oeis.org

11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mitch Harris

Cf. A109004.

GCD[Range[0,100],11] (* Harvey P. Dale, May 14 2022 *)

from math import gcd
def a(n): return gcd(n, 11)
print([a(n) for n in range(99)]) # Michael S. Branicky, Nov 01 2021

a(n) = 1 + 10*[11|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-11).
Multiplicative with a(p^e, 11) = gcd(p^e, 11). - David W. Wilson, Jun 12 2005
Dirichlet g.f.: zeta(s)*(1+10/11^s). - R. J. Mathar, Apr 08 2011
a(n) = ((n-1) mod 2 + 1)*(10*floor(((n-1) mod 11)/10) + 1). - Gary Detlefs, Dec 28 2011

A109015 a(n) = gcd(n,12).

Original entry on oeis.org

12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2

Offset: 0

Mitch Harris

Periodic, with period = 12. - Harvey P. Dale, Dec 20 2018

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

Cf. A109004.

GCD[Range[0,100],12] (* or *) PadRight[{},120,{12,1,2,3,4,1,6,1,4,3,2,1}] (* Harvey P. Dale, Dec 20 2018 *)

from math import gcd
def a(n): return gcd(n, 12)
print([a(n) for n in range(99)]) # Michael S. Branicky, Dec 01 2021

a(n) = 1 + [2|n] + 2*[3|n] + 2*[4|n] + 2*[6|n] + 4*[12|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-12).
Multiplicative with a(p^e, 12) = gcd(p^e, 12). - David W. Wilson, Jun 12 2005
Dirichlet g.f.: zeta(s)*(1 + 1/2^s + 2/4^s)*(1 + 2/3^s). - R. J. Mathar, Apr 08 2011

A159335 Triangle read by rows: numerator of n/binomial(n,m).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 2, 1, 4, 5, 1, 1, 1, 1, 5, 6, 1, 2, 3, 2, 1, 6, 7, 1, 1, 1, 1, 1, 1, 7, 8, 1, 2, 1, 4, 1, 2, 1, 8, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 10, 1, 2, 1, 1, 5, 1, 1, 2, 1, 10, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 2, 3, 4, 1, 1, 1, 4, 3, 2, 1, 12, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 0

Leroy Quet, Apr 10 2009

This triangle first differs from A109004 (read as a triangle) at T(10, 4) and T(10,6).

T(n,m) is the smallest positive integer such that binomial(n,m)*T(n,m) is a multiple of n.

			Row 10 of Pascal's triangle is: 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. {a(10,m)} of this sequence (A159335) is: 10, 1, 2, 1, 1, 5, 1, 1, 2, 1,10. Multiplying the corresponding integers, we get multiples of 10: 1*10=10,10*1=10, 45*2=90, 120*1=120, 210*1=210, 252*5=1260, 210*1=210, 120*1=120, 45*2=90, 10*1=10, 1*10=10.

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A165661 (denominators), A007318, A020475, A109004.

/* As triangle */ [[n/GCD(n,Binomial(n, k)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 25 2018

Table[n/GCD[n, Binomial[n, k]], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Jun 25 2018 *)

for(n=0, 10, for(k=0,n, print1(n/gcd(n, binomial(n,k)), ", "))) \\ G. C. Greubel, Jun 25 2018

T(n,m) = n/gcd(n,binomial(n,m)).

Extended by Ray Chandler, Jun 19 2009

Edited by Franklin T. Adams-Watters, Sep 24 2009

A165661 Triangle read by rows: denominator of n/binomial(n,m).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 10, 5, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 1, 9, 12, 21, 126, 21, 12, 9, 1, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 1, 11, 55, 165, 66, 77, 66, 165, 55, 11, 1, 1

Offset: 0

Franklin T. Adams-Watters, Sep 24 2009

First difference from A107711 is at T(10,4) = 21 instead of 42.

Cf. A159335 (numerators), A107711, A007318, A109004.

T(n,m) = binomial(n,m)/gcd(n,binomial(n,m)).

A281726 Triangular array read by rows: T(n,k) is the number of elements in an n X k matrix that will be assigned the same value whether the integers from 1 to n*k are assigned to elements in row-major order or column-major order.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 2, 2, 4, 5, 2, 3, 2, 5, 6, 2, 2, 2, 2, 6, 7, 2, 3, 4, 3, 2, 7, 8, 2, 2, 2, 2, 2, 2, 8, 9, 2, 3, 2, 5, 2, 3, 2, 9, 10, 2, 2, 4, 2, 2, 4, 2, 2, 10, 11, 2, 3, 2, 3, 6, 3, 2, 3, 2, 11, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 12, 13, 2, 3, 4, 5, 2, 7, 2, 5, 4, 3, 2, 13

Offset: 1

Michel Marcus, Jan 28 2017

From Jon E. Schoenfield, Dec 10 2023: (Start)
T(n,k) is also the number of lattice points that lie on a line segment from (0,0) to (n-k,k-1). Thus, row n of the triangle lists, for each of the n 1st-quadrant lattice points P whose Manhattan distance from the origin is n-1, the number of lattice points on a line segment from the origin to P.
E.g., for n = 5, the 5 1st-quadrant lattice points whose Manhattan distance from the origin is 4 are (0,4), (1,3), (2,2), (3,1), and (4,0), and a line segment drawn from the origin to each of these points will intersect 5, 2, 3, 2, and 5 lattice points, respectively; { 5, 2, 3, 2, 5 } is row 5 of the triangle. (End)

			For n=3 and k=2, the matrix will be
  1 2  and  1 4
  3 4       2 5
  5 6       3 6
and there are 2 identical terms (1 and 6).
Triangle begins:
  1;
  2, 2;
  3, 2, 3;
  4, 2, 2, 4;
  5, 2, 3, 2, 5;
  6, 2, 2, 2, 2, 6;
  ...

Alois P. Heinz, Rows n = 1..200, flattened
Ana Rechtman, 3ème Défi du Calendrier Mathématique, Images des Mathématiques, CNRS, January 2017.

Cf. A109004, A281725.

Main diagonal and column k=1 give A000027.

T:= (n, k)-> add(add(`if`(j+k*(i-1)=i+n*(j-1), 1, 0), i=1..n), j=1..k):
seq(seq(T(n,k), k=1..n), n=1..20);  # Alois P. Heinz, Jan 28 2017

Array[1+GCD[#,Range[0,#]]&,20,0] (* Paolo Xausa, Dec 08 2023 *)

a(n, k) = {ml = matrix(n, k, i, j, ((i-1)*k+j)); mc = matrix(n, k, i, j, ((j-1)*n+i)); sum(i=1, n, sum(j=1, k, ml[i,j] == mc[i,j]));}

T(n,k) = 1 + gcd(n-1, k-1). - Jon E. Schoenfield, Dec 08 2023

A140682 Triangle T(n,k) = gcd(n,k)-binomial(n,k), 0<=k<=n.

Original entry on oeis.org

-1, 0, 0, 1, -1, 1, 2, -2, -2, 2, 3, -3, -4, -3, 3, 4, -4, -9, -9, -4, 4, 5, -5, -13, -17, -13, -5, 5, 6, -6, -20, -34, -34, -20, -6, 6, 7, -7, -26, -55, -66, -55, -26, -7, 7, 8, -8, -35, -81, -125, -125, -81, -35, -8, 8, 9, -9, -43, -119, -208, -247, -208, -119, -43, -9, 9

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 11 2008

Row sums are -1, 0, 1, 0, -4, -18, -43, -108, -228, -482, -987...

			-1;
0, 0;
1, -1, 1;
2, -2, -2, 2;
3, -3, -4, -3, 3;
4, -4, -9, -9, -4, 4;
5, -5, -13, -17, -13, -5, 5;
6, -6, -20, -34, -34, -20, -6, 6;
7, -7, -26, -55, -66, -55, -26, -7, 7;
8, -8, -35, -81, -125, -125, -81, -35, -8, 8;
9, -9, -43, -119, -208, -247, -208, -119, -43, -9, 9;

Cf. A109004.

A140682 := proc(n,k)
    igcd(n,k)-binomial(n,k) ;
end proc: # R. J. Mathar, Jan 17 2013

Clear[p, x, n] p[x_, n_] = Sum[(GCD[n, i] - Binomial[n, i])*x^i, {i, 0, n}]; Table[ExpandAll[p[x, n]], {n, 1, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[a]

T(n,k) = T(n,n-k).

T(n,k) = A109004(n,k)-A007318(n,k). - R. J. Mathar, Jan 17 2013

New name, editing, and missing leading terms added. - R. J. Mathar, Jan 17 2013

A266685 T(n,k) is the number of loops appearing in pattern of circular arc connecting two vertices of regular polygons. (See Comments.)

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 3, 2, 1, 6, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 18

Offset: 0

Kival Ngaokrajang, Jan 02 2016

The patterns in A262343 and A264906 can be considered as case of skip 0 and 1 vertex of circle construction on regular polygons. k is the cyclic number of loops of the case skip n-vertices. See illustration for more details.

T(n,k) is conjectured to be even rows of A109004 (excluding the first column).

			Irregular triangle begins:
n\k   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
0     1  2
1     1  2  1  4
2     1  2  3  2  1  6
3     1  2  1  4  1  2  1  8
4     1  2  1  2  5  2  1  2  1 10
5     1  2  3  4  1  6  1  4  3  2  1 12
6     1  2  1  2  1  2  7  2  1  2  1  2  1 14
7     1  2  1  4  1  2  1  8  1  2  1  4  1  2  1 16
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A109004, A262343, A264906.

Table[GCD[2 n + 3 + k, k + 1], {n, 0, 8}, {k, 0, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 03 2016 *)

for (n=0, 20,for (k=0, 2*n+2, t=gcd(2*n+3+k, k+1); print1(t, ", ")))

T(n,k) = gcd(2*n+3+k, k+1), n >= 0, k = 0..2*n+1.

cp's OEIS Frontend

A109013 a(n) = gcd(n,10).

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Mathematica

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A109010 a(n) = gcd(n,7).

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Maple

Mathematica

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A109011 a(n) = gcd(n,8).

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Mathematica

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A109014 a(n) = gcd(n,11).

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Mathematica

Python

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A109015 a(n) = gcd(n,12).

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A159335 Triangle read by rows: numerator of n/binomial(n,m).

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Magma

Mathematica

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A165661 Triangle read by rows: denominator of n/binomial(n,m).

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A281726 Triangular array read by rows: T(n,k) is the number of elements in an n X k matrix that will be assigned the same value whether the integers from 1 to n*k are assigned to elements in row-major order or column-major order.

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Maple