A010875 -id:A010875 - cp's OEIS frontend

A099547 Odd part of n modulo 6.

1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 5, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 5, 3, 5, 5, 3, 1, 1, 3, 1, 5, 3

Offset: 1

Ralf Stephan, Oct 23 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000265, A010875.

Cf. A099544, A099545, A099546, A099548, A099549, A099550, A099551.

a[n_] := Mod[n / 2^IntegerExponent[n, 2], 6]; Array[a, 100] (* Amiram Eldar, Aug 29 2024 *)

a(n) = (n>>valuation(n, 2))%6 \\Charles R Greathouse IV, May 14 2014

a(n) = A010875(A000265(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. - Amiram Eldar, Aug 29 2024

A130488 a(n) = Sum_{k=0..n} (k mod 10) (Partial sums of A010879).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 90, 91, 93, 96, 100, 105, 111, 118, 126, 135, 135, 136, 138, 141, 145, 150, 156, 163, 171, 180, 180, 181, 183, 186, 190, 195, 201, 208, 216, 225, 225, 226, 228, 231, 235, 240, 246, 253

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 10, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

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

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A130481, A130482, A130483, A130484, A130485, A130486, A130487.

a:=[0,1,3,6,10,15,21,28,36,45,45];; for n in [12..61] do a[n]:=a[n-1]+a[n-10]-a[n-11]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,36,45,45]; [n le 11 select I[n] else Self(n-1) + Self(n-10) - Self(n-11): n in [1..61]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-10*x^9+9*x^10)/((1-x^10)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Aug 31 2019

LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1}, {0,1,3,6,10,15,21,28,36,45, 45}, 60] (* G. C. Greubel, Aug 31 2019 *)

a(n) = sum(k=0, n, k % 10); \\ Michel Marcus, Apr 28 2018

def A130488(n):
    a, b = divmod(n,10)
    return 45*a+(b*(b+1)>>1) # Chai Wah Wu, Jul 27 2022

def A130488_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-10*x^9+9*x^10)/((1-x^10)*(1-x)^3)).list()
A130488_list(60) # G. C. Greubel, Aug 31 2019

a(n) = 45*floor(n/10) + A010879(n)*(A010879(n) + 1)/2.
G.f.: (Sum_{k=1..9} k*x^k)/((1-x^10)*(1-x)).
G.f.: x*(1 - 10*x^9 + 9*x^10)/((1-x^10)*(1-x)^3).

A084300 a(n) = phi(n) mod 6.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 02 2003

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A010875, A054763, A074942, A084299, A084301.

a[n_] := Mod[EulerPhi[n], 6]; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *)

A084300(n) = (eulerphi(n)%6); \\ Antti Karttunen, Aug 22 2017

a(n) = A000010(n) mod 6.

a(n) = A010875(A000010(n)). - Amiram Eldar, Aug 17 2024

A277544 a(n) = n/6^m mod 6, where 6^m is the greatest power of 6 that divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 4, 1, 2, 3, 4, 5, 5, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 4, 1, 2, 3, 4, 5, 5, 1, 2, 3, 4, 5, 2, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 2, 1, 2

Offset: 1

Clark Kimberling, Oct 19 2016

a(n) is the rightmost nonzero digit in the base 6 expansion of n.

			a(8) = (8/6 mod 6) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A010875, A244414.

Table[Mod[n/6^IntegerExponent[n, 6], 6], {n, 1, 160}]

a(n) = n/6^valuation(n, 6) % 6; \\ Michel Marcus, Oct 20 2016

a(n) = A244414(n) mod 6. - Michel Marcus, Oct 20 2016

A010887 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130486(n)+n+1. - Hieronymus Fischer, Jun 08 2007

1371742/11111111 = 0.123456781234567812345678... - Eric Desbiaux, Nov 03 2008

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

Cf. A010872, A010873, A010874, A010875, A010876, A010878, A004526, A002264, A002265, A002266.

Cf. A177034 (decimal expansion of (9280+3*sqrt(13493990))/14165). - Klaus Brockhaus, May 01 2010

a010887 = (+ 1) . flip mod 8
a010887_list = cycle [1..8]
-- Reinhard Zumkeller, Nov 09 2014, Mar 04 2014

PadRight[{},90,Range[8]] (* Harvey P. Dale, May 10 2022 *)

def A010887(n): return 1 + (n & 7) # Chai Wah Wu, May 25 2022

a(n) = 1 + (n mod 8) - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = (1/2)*(9 - (-1)^n - 2*(-1)^(b/4) - 4*(-1)^((b - 2 + 2*(-1)^(b/4))/8)) where b = 2n - 1 + (-1)^n.
Also a(n) = A010877(n) + 1.
G.f.: g(x) = (1/(1-x^8))*Sum_{k=0..7} (k+1)*x^k.
Also: g(x) = (8x^9 - 9x^8 + 1)/((1-x^8)*(1-x)^2). (End)

A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 8, 9, 13, 12, 17, 18, 22, 24, 30, 30, 38, 40, 46, 50, 59, 60, 71, 75, 84, 90, 102, 105, 120, 126, 138, 147, 163, 168, 187, 196, 212, 224, 244, 252, 276, 288, 308, 324, 349, 360, 389, 405, 430, 450, 480, 495, 530, 550, 580, 605, 641, 660, 701, 726

Offset: 0

N. J. A. Sloane, Dec 30 2003

nonn

Poincaré series [or Poincare series] (or Molien series) for (P[x_0,x_1] ⊗ P[x_0,x_1])^(S_2).

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 199.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1).

Cf. A089599, A091434, A091726, A091769.

Cf. A010875 (n mod 6). Contains A006002 and A212683. - Luce ETIENNE, Aug 14 2018

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (1-x^10)/(&*[1-x^j: j in [2..6]]) )); // G. C. Greubel, Apr 02 2023

seq(coeff(series((1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), x,n+1),x,n),n=0..70); # Muniru A Asiru, Aug 15 2018

CoefficientList[Series[(1-x^10)/Product[1-x^(j+1), {j,5}], {x,0,70}], x] (* G. C. Greubel, Apr 02 2023 *)

Vec((1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)) + O(x^100)) \\ Michel Marcus, Mar 19 2014

def A082146_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^10)/prod(1-x^j for j in range(2,7)) ).list()
A082146_list(70) # G. C. Greubel, Apr 02 2023

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-11) + a(n-13) - a(n-14).
G.f.: ( 1+x^2+x^4-x-x^3 ) / ( (1+x^2)*(1-x+x^2)*(1+x)^2*(1+x+x^2)^2*(1-x)^4 ). - R. J. Mathar, Oct 11 2011
a(n) = (120*floor(n/6)^3 + 60*(m+5)*floor(n/6)^2 - 20*(m^5-13*m^4 +60*m^3-116*m^2+74*m-18)*floor(n/6) - (19*m^5-245*m^4+1125*m^3-2185*m^2+1496*m-210) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 33, 34, 17, 10, 5, 35, 7, 12, 36, 37, 24, 22, 7, 16, 13, 38, 5, 39, 7, 12, 40, 21, 41, 27, 7, 42, 43, 10, 5, 44, 33, 12, 13, 26, 5, 45, 46, 16, 17, 10, 24, 47, 7, 48, 28, 49, 5, 22, 7, 34

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010875(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Many of the same comments as given in A319717 apply also here, except for this filter, the "blind spot" area (where only unique values are possible for a(n)) is different, and contains at least all numbers in A070003. Because presence of 2 or 3 in the prime factorization of n do not force the value of a(n) unique, this is substantially less lax (i.e., more exact) filter than A319717. Here among the first 100000 terms, only 2393 have a unique value, compared to 74355 in A319717.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319690(i) = A319690(j).

			For n = 15 (3*5) and n = 33 (3*11), the mod 6 residue of the largest prime factor is 5, also in both cases it is unitary (A319988(n) = 1), and the quotient n/A006530(n) is equal, in this case 3. Thus a(15) and a(33) are alloted the same running count (13 in this case) by rgs-transform.
For n = 2275 (5^2 * 7 * 13), n = 3325 (5^2 * 7 * 19), 5425 (5^2 * 7 * 31) and 6475 (5^2 * 7 * 37), the largest prime factor = 1 (mod 6), and A052126(n) = 175, thus these numbers are allotted the same running count (394 in this case) by rgs-transform.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A006530, A052126, A319988, A319717.
Cf. A007528 (positions of 5's), A002476 (of 7's), A112774 (after its initial term gives the position of 10's in this sequence).
Cf. also A319994 (modulo 4 analog).

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319996aux(n) = if(1==n,0,[A006530(n)%6, A052126(n), A319988(n)]);
v319996 = rgs_transform(vector(up_to,n,A319996aux(n)));
A319996(n) = v319996[n];

A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jun 12 2001

			T(5,3)=5^(3-1) mod 3=25 mod 3=1. Rows start (0,1,1,1,1,...), (0,0,1,0,1,...), (0,1,0,3,1...), (0,0,1,0,1,...), (0,1,1,1,0,...), ...

Cf. A002997, A060154. Rows include A057427, A062173, A062174, A062175, A062176. Columns include A000004, A000035, A011655, A010684 with interleaved 0's, A011558, A010875. Diagonals include all the rows again and A000004 and A009001 unsigned.

A130489 a(n) = Sum_{k=0..n} (k mod 11) (Partial sums of A010880).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110, 110, 111, 113, 116, 120, 125, 131, 138, 146, 155, 165, 165, 166, 168, 171, 175, 180, 186, 193, 201, 210, 220, 220, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 275, 276

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 11, A[i,i]:=1, A[i,i-1]=-1. Then, for n >= 1, a(n)=det(A). - Milan Janjic, Jan 24 2010

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

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488.

a:=[0,1,3,6,10,15,21,28,36,45, 55,55];; for n in [13..61] do a[n]:=a[n-1]+a[n-11]-a[n-12]; od; a; # G. C. Greubel, Aug 31 2019

I:=[0,1,3,6,10,15,21,28,36,45,55,55]; [n le 12 select I[n] else Self(n-1) + Self(n-11) - Self(n-12): n in [1..61]]; // G. C. Greubel, Aug 31 2019

seq(coeff(series(x*(1-11*x^10+10*x^11)/((1-x^11)*(1-x)^3), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Aug 31 2019

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1}, {0,1,3,6,10,15,21,28,36,45, 55,55}, 60] (* G. C. Greubel, Aug 31 2019 *)
Accumulate[PadRight[{},80,Range[0,10]]] (* Harvey P. Dale, Jul 21 2021 *)

a(n) = sum(k=0, n, k % 11); \\ Michel Marcus, Apr 28 2018

def A130489_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-11*x^10+10*x^11)/((1-x^11)*(1-x)^3)).list()
A130489_list(60) # G. C. Greubel, Aug 31 2019

a(n) = 55*floor(n/11) + A010880(n)*(A010880(n) + 1)/2.
G.f.: (Sum_{k=1..10} k*x^k)/((1-x^11)*(1-x)).
G.f.: x*(1 - 11*x^10 + 10*x^11)/((1-x^11)*(1-x)^3).

A010885 Period 6: repeat [1, 2, 3, 4, 5, 6].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3

Offset: 0

N. J. A. Sloane

Partial sums are given by A130484(n)+n+1. - Hieronymus Fischer, Jun 08 2007

41152/333333 = 0.123456123456123456... [Eric Desbiaux, Nov 03 2008]

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

Cf. A010872, A010873, A010874, A010875, A010876, A004526, A002264, A002265, A002266.

Cf. A177158 (decimal expansion of (103+2*sqrt(4171))/162). [From Klaus Brockhaus, May 03 2010]

&cat[[1..6]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A010885:=n->(21-3*cos(n*Pi)-4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/6: seq(A010885(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[Range[6],{n,15}]] (* Harvey P. Dale, Aug 01 2011 *)
PadRight[{},90,Range[6]] (* Harvey P. Dale, Sep 23 2019 *)

a(n) = 1 + (n mod 6). - Paolo P. Lava, Nov 21 2006
a(n) = A010875(n)+1. G.f.: g(x)=(Sum_{0<=k<6} (k+1)*x^k)/(1-x^6). Also g(x)=(6*x^7-7*x^6+1)/((1-x^6)*(1-x)^2). - Hieronymus Fischer, Jun 08 2007
From Wesley Ivan Hurt, Jun 17 2016: (Start)
G.f.: (1+2*x+3*x^2+4*x^3+5*x^4+6*x^5)/(1-x^6).
a(n) = (21-3*cos(n*Pi)-4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/6.
a(n) = a(n-6) for n>5. (End)

cp's OEIS Frontend

A099547 Odd part of n modulo 6.

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A130488 a(n) = Sum_{k=0..n} (k mod 10) (Partial sums of A010879).

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A084300 a(n) = phi(n) mod 6.

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A277544 a(n) = n/6^m mod 6, where 6^m is the greatest power of 6 that divides n.

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A010887 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8.

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A082146 Expansion of g.f.: (1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).

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A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

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A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).