A010878 -id:A010878 - cp's OEIS frontend

A099550 Odd part of n modulo 9.

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

Offset: 1

Ralf Stephan, Oct 23 2004

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

Cf. A000265, A010878.

Cf. A099544, A099545, A099546, A099547, A099548, A099549, A099551.

a[n_] := Mod[n / 2^IntegerExponent[n, 2], 9]; Array[a, 100] (* Amiram Eldar, Aug 29 2024 *)

a(n) = (n>>valuation(n, 2))%9 \\Charles R Greathouse IV, May 14 2014

a(n) = A010878(A000265(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - 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).

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)

A070433 a(n) = n^2 mod 9.

Original entry on oeis.org

0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1

Offset: 0

N. J. A. Sloane, May 12 2002

Also decimal expansion of 4692347/333333333. - Enrique Pérez Herrero, Nov 27 2022

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

Cf. A000290, A010878, A053879, A070430, A070431.

Table[Mod[n^2,9],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2011 *)
PowerMod[Range[0,200],2,9] (* Harvey P. Dale, Jun 11 2011 *)

a(n)=[0,1,4,0,7,7,0,4,1][n%9+1] \\ Charles R Greathouse IV, Jun 11 2011

a(n)=n^2%9 \\ Charles R Greathouse IV, Jun 11 2011

From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-9).
G.f.: ( -x*(1+x)*(x^6+3*x^5-3*x^4+10*x^3-3*x^2+3*x+1) ) / ( (x-1)*(1+x+x^2)*(x^6+x^3+1) ). (End)
a(n) = A010878(A000290(n)) = A010878(n^2). - Enrique Pérez Herrero, Nov 27 2022

A177274 Periodic sequence: Repeat 1, 2, 3, 4, 5, 6, 7, 8, 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6

Offset: 0

Klaus Brockhaus, May 07 2010

Interleaving of A131669 and A131669 without first five terms.
Continued fraction expansion of (684125+sqrt(635918528029))/1033802.
Decimal expansion of 13717421/111111111.
a(n) = A010888(n+1) = A010878(n)+1 = A117230(n+2)-1.
a(n) = A064806(n+1)-n-1.
Essentially first differences of A037123.

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

Cf. A131669 (odd digits followed by positive even digits), A010888 (digital root of n), A010878 (n mod 9), A117230 (1 followed by (repeat 2, 3, 4, 5, 6, 7, 8, 9, 10), offset 1), A064806 (n + digital root of n), A037123, A177270 (decimal expansion of (684125+sqrt(635918528029))/1033802).

&cat[ [1, 2, 3, 4, 5, 6, 7, 8, 9]: k in [1..12] ];

PadRight[{},120,Range[9]] (* Paolo Xausa, Jan 08 2024 *)

a(n) = (n mod 9)+1.
a(n) = a(n-9) for n > 8; 1; a(n) = n+1 for n <= 8.
G.f.: (1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8)/(1-x^9). [corrected by Georg Fischer, May 11 2019]

A105852 a(n) = sigma(n) mod 9.

Original entry on oeis.org

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

Offset: 1

Shyam Sunder Gupta, May 05 2005

If gcd(m,n) = 1 then a(m*n) = (a(m) * a(n)) mod 9. - Robert Israel, Sep 14 2014

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

Cf. A000203, A010878, A190998 (digital root of sigma(n)).

seq(numtheory:-sigma(n) mod 9, n=1..1000); # Robert Israel, Sep 14 2014

Mod[DivisorSigma[1, Range[100]], 9] (* Wesley Ivan Hurt, Apr 25 2023 *)

PARI
```
a(n)=sigma(n)%9
```

a(n) = A010878(A000203(n)). - Michel Marcus, Sep 14 2014

Name corrected and keyword base removed by Michel Marcus, Sep 14 2014

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).

A010889 Simple periodic sequence: repeat 1,2,3,4,5,6,7,8,9,10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1

Offset: 0

N. J. A. Sloane

Partial sums are given by A130488(n)+n+1. - Hieronymus Fischer, Jun 08 2007

Continued fraction expansion of (232405+sqrt(71216963807))/348378. [From Klaus Brockhaus, May 15 2010]

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

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A004526, A002264, A002265, A002266.

Cf. A177933 (decimal expansion of (232405+sqrt(71216963807))/348378). [From Klaus Brockhaus, May 15 2010]

PadRight[{},120,Range[10]] (* Harvey P. Dale, Feb 22 2015 *)

def a(n): return n % 10 + 1 # Paul Muljadi, Aug 06 2024

a(n) = 1 + (n mod 10) - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
a(n) = A010879(n)+1.
G.f.: (Sum_{k=0..9} (k+1)*x^k)/(1-x^10).
G.f.: (10x^11-11x^10+1)/((1-x^10)(1-x)^2). (End)

More terms from Klaus Brockhaus, May 15 2010

A014018 Inverse of 9th cyclotomic polynomial.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0

Offset: 0

Simon Plouffe

Periodic with period length 9. - Ray Chandler, Apr 03 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, -1, 0, 0, -1).
Index to sequences related to inverse of cyclotomic polynomials

Cf. A010878.

&cat[[1,0,0,-1,0,0,0,0,0]: n in [0..15]]; // Vincenzo Librandi, Apr 03 2014

with(numtheory,cyclotomic); c := n->series(1/cyclotomic(n,x),x,80);

CoefficientList[Series[1/Cyclotomic[9, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{0, 0, -1, 0, 0, -1},{1, 0, 0, -1, 0, 0},81] (* Ray Chandler, Sep 15 2015 *)

Vec(1/polcyclo(9)+O(x^99)) \\ Charles R Greathouse IV, Mar 24 2014

G.f.: 1/(1 + x^3 + x^6). - Ilya Gutkovskiy, Aug 18 2017

a(n) = (19*m^8 - 628*m^7 + 8526*m^6 - 61152*m^5 + 247611*m^4 - 558012*m^3 + 637604*m^2 - 287408*m + 13440)/13440, where m = n mod 9. - Luce ETIENNE, Oct 18 2018

A086360 The n-th primorial number reduced modulo 9.

Original entry on oeis.org

1, 2, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 3, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 6, 6, 6, 6, 3, 6, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 3, 3, 3, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6

Offset: 0

Labos Elemer, Jul 21 2003

a(n) is the fixed point reached by decimal-digit-sum-function (A007953), when starting the iteration from the value of the n-th primorial, A002110(n). - The (edited) original definition of the sequence, which is equal to a simple definition a(n) = A002110(n) mod 9, because taking the decimal digit sum preserves congruence modulo 9. - Antti Karttunen, Nov 14 2024

Only a(0)=1 and a(1)=2; each subsequent term is either a 3 or a 6.

			For n=7, 7th primorial = 510510, list of iterated digit sums is {510510,12,3}, thus a(7)=3.

Antti Karttunen, Table of n, a(n) for n = 0..19683 (terms 1..10000 from Nathaniel Johnston)
Index entries for sequences related to primorial numbers

Cf. A002110, A007953, A010878, A010888, A029898, A038194, A086353-A086361.

Cf. also A377876, A377877.

A086360 := proc(n) option remember: if(n=1)then return 2:fi: return ithprime(n)*procname(n-1) mod 9: end: seq(A086360(n), n=1..100); # Nathaniel Johnston, May 04 2011

sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[sud, q[w]], {w, 1, 128}]

up_to = 19683;
A086360list(up_to_n) = { my(m=9, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr = (pr*prime(n))%m; v[1+n] = pr); (v); };
v086360 = A086360list(up_to);
A086360(n) = v086360[1+n]; \\ Antti Karttunen, Nov 14 2024

a(n) = A010878(A002110(n)) = A002110(n) mod 9.

a(n) = A010888(A002110(n)).

Term a(0)=1 prepended, old definition moved to comments and replaced with one of the formulas, keyword:base removed because not really base-dependent - Antti Karttunen, Nov 14 2024

cp's OEIS Frontend

A099550 Odd part of n modulo 9.

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

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

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A070433 a(n) = n^2 mod 9.

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A177274 Periodic sequence: Repeat 1, 2, 3, 4, 5, 6, 7, 8, 9.

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A105852 a(n) = sigma(n) mod 9.

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

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