A010875 -id:A010875 - cp's OEIS frontend

A010889 Simple periodic sequence: repeat 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, 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

A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 66, 67, 69, 72, 76, 81, 87, 94, 102, 111, 121, 132, 132, 133, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 198, 199, 201, 204, 208, 213, 219, 226, 234, 243, 253, 264, 264, 265, 267, 270, 274, 279, 285, 292, 300

Offset: 0

Hieronymus Fischer, May 31 2007

nonn

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 12, 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 (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

List([0..60], n-> Sum([0..n], k-> k mod 12 )); # G. C. Greubel, Sep 01 2019

[&+[(k mod 12): k in [0..n]]: n in [0..60]]; // G. C. Greubel, Sep 01 2019

seq(coeff(series(x*(1-12*x^11+11*x^12)/((1-x^12)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 01 2019

Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,3,6,10,15,21,28,36,45,55,66,66},60] (* Harvey P. Dale, Jan 16 2024 *)

a(n) = sum(k=0, n, k % 12); \\ Michel Marcus, Apr 29 2018

[sum(k%12 for k in (0..n)) for n in (0..60)] # G. C. Greubel, Sep 01 2019

a(n) = 66*floor(n/12) + A010881(n)*(A010881(n) + 1)/2.
G.f.: (Sum_{k=1..11} k*x^k)/((1-x^12)*(1-x)).
G.f.: x*(1 - 12*x^11 + 11*x^12)/((1-x^12)*(1-x)^3).

A204671 a(n) = n^n (mod 6).

Original entry on oeis.org

1, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4, 3, 4, 5, 0, 1, 4

Offset: 0

José María Grau Ribas, Jan 18 2012

For n>0, periodic with period 6 = A174824: repeat [1, 4, 3, 4, 5, 0].

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

Cf. A000312, A010875, A174824, A204690.

[1] cat &cat [[1, 4, 3, 4, 5, 0]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A204671:=n->[1, 4, 3, 4, 5, 0][(n mod 6)+1]: 1, seq(A204671(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

Table[PowerMod[n,n,6], {n,0,140}]
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 1},{1, 4, 3, 4, 5, 0},86]] (* Ray Chandler, Aug 26 2015 *)

a(n)=lift(Mod(n, 6)^n) \\ Andrew Howroyd, Feb 25 2018

G.f.: (x^6-5*x^5-4*x^4-3*x^3-4*x^2-x-1)/((x-1)*(x+1)*(x^2-x+1)*(x^2+x+1)). [Colin Barker, Jul 20 2012]
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(0) = 1, a(n) = (17 - cos(n*Pi) - 8*cos(n*Pi/3) - 8*cos(2*n*Pi/3) - 4*sqrt(3)*sin(n*Pi/3) - 4*sqrt(3)*sin(2*n*Pi/3))/6 for n>0. (End)
a(n) = A010875(A000312(n)). - Michel Marcus, Jun 27 2016

A319707 Filter sequence which records for primes their residue modulo 6, and for all other numbers assigns a unique number.

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, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A010875(n) when n is a prime, otherwise -n.
Primes of the form 6k+5 (A007528) get value 5, and the primes of the form 6k+1 (A002476) get value 7, while for all other n, a(n) is assigned to a unique running count.
For all i, j:
  a(i) = a(j) => A010875(i) = A010875(j),
  a(i) = a(j) => A305900(i) = A305900(j),
  a(i) = a(j) => A319717(i) = A319717(j) => A319716(i) = A319716(j).

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

Cf. A319716, A319717.
Cf. A007528 (positions of 5's), A002476 (positions of 7's).
Cf. also A319704.
Differs from A319716 for the first time at n=121.

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; };
A319707aux(n) = if(isprime(n),(n%6),-n);
v319707 = rgs_transform(vector(up_to,n,A319707aux(n)));
A319707(n) = v319707[n];

A008670 Molien series for Weyl group F_4.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 7, 9, 11, 12, 16, 18, 20, 24, 28, 30, 36, 40, 44, 50, 56, 60, 69, 75, 81, 90, 99, 105, 117, 126, 135, 147, 159, 168, 184, 196, 208, 224, 240, 252, 272, 288, 304, 324, 344, 360, 385, 405, 425, 450, 475, 495, 525, 550, 575, 605, 635, 660, 696, 726, 756

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 3, 4 and 6. - Ilya Gutkovskiy, May 24 2017

Coxeter and Moser, Generators and Relations for Discrete Groups, Table 10.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 28).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 236
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1).

Cf. A002411, A006002, A010875, A011934, A027480, A055232, A182260.

MolienSeries(CoxeterGroup("F4")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 08 2019

a:= proc(n) local m, r; m := iquo (n, 12, 'r'); r:= r+1; ([4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26][r]+ (6+r+4*m)*m)*m+ [1$3, 2, 3$2, 5, 6, 7, 9, 11, 12][r] end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 06 2008

Take[CoefficientList[Series[1/((1-x^2)(1-x^6)(1-x^8)(1-x^12)),{x,0,130}], x], {1,-1,2}] (* or *) LinearRecurrence[ {1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1},{1,1,1,2,3,3,5,6,7,9,11,12,16,18},70] (* Harvey P. Dale, Feb 07 2012 *)

my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))) \\ G. C. Greubel, Sep 08 2019

def A008670_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))).list()
A008670_list(70) # G. C. Greubel, Sep 08 2019

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)). [Corrected by Ralf Stephan, Apr 29 2014]
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), with a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(6)=5, a(7)=6, a(8)=7, a(9)=9, a(10)=11, a(11)=12, a(12)=16, a(13)=18. - Harvey P. Dale, Feb 07 2012
a(n) ~ (1/432)*n^3. - Ralf Stephan, Apr 29 2014
a(n) = (120*floor(n/6)^3 + 60*(m+7)*floor(n/6)^2 + 2*(m^5-15*m^4+75*m^3-135*m^2+134*m+240)*floor(n/6) + 3*(m^5-15*m^4+75*m^3-135*m^2+84*m+70) + (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
a(n) = 1 + floor((2*n^3 + 42*n^2 + n*(279 + 9*(-1)^n - 48*[(n mod 3)=2]))/864) where [] is the Iverson bracket. - Hoang Xuan Thanh, Jun 22 2025

A010884 Period 5: repeat [1,2,3,4,5].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Partial sums are given by A130483(n)+n+1. - Hieronymus Fischer, Jun 08 2007

4115/33333 = 0.12345123451234512345... - Eric Desbiaux, Nov 03 2008

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

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

Cf. A177038 (decimal expansion of (195+sqrt(65029))/314).

PadRight[{},120,Range[5]] (* Harvey P. Dale, Dec 08 2018 *)

a(n)=n%5+1 \\ Charles R Greathouse IV, Jul 13 2016

a(n) = 1 + (n mod 5). - Paolo P. Lava, Nov 21 2006
From Hieronymus Fischer, Jun 08 2007: (Start)
G.f.: (5*x^4+4*x^3+3*x^2+2*x+1)/(1-x^5) = (5*x^6-6*x^5+1)/((1-x^5)*(1-x)^2).
a(n) = A010874(n)+1. (End)
a(n) = a(n-5). - Wesley Ivan Hurt, Jan 15 2022

A053841 (Sum of digits of n written in base 6) modulo 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 0, 2, 3, 4, 5, 0, 1, 3, 4, 5, 0, 1, 2, 4, 5, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 0, 2, 3, 4, 5, 0, 1, 3, 4, 5, 0, 1, 2, 4, 5, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 0, 1, 3, 4, 5, 0, 1, 2, 4, 5, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 1, 2, 3

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the fifth row of the array in A141803. - Andrey Zabolotskiy, May 18 2016

Paolo Xausa, Table of n, a(n) for n = 0..10000
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010875, A053827, A141803.

Cf. A010060, A053837-A053844.

Mod[DigitSum[Range[0, 100], 6], 6] (* Paolo Xausa, Aug 09 2024 *)

a(n) = vecsum(digits(n, 6)) % 6; \\ Michel Marcus, May 18 2016

a(n) = A010875(A053827(n)). - Andrey Zabolotskiy, May 18 2016

A080063 a(n) = n mod (spf(n)+1), where spf(n) is the smallest prime dividing n (A020639).

Original entry on oeis.org

1, 2, 3, 1, 5, 0, 7, 2, 1, 1, 11, 0, 13, 2, 3, 1, 17, 0, 19, 2, 1, 1, 23, 0, 1, 2, 3, 1, 29, 0, 31, 2, 1, 1, 5, 0, 37, 2, 3, 1, 41, 0, 43, 2, 1, 1, 47, 0, 1, 2, 3, 1, 53, 0, 1, 2, 1, 1, 59, 0, 61, 2, 3, 1, 5, 0, 67, 2, 1, 1, 71, 0, 73, 2, 3, 1, 5, 0, 79, 2, 1, 1, 83, 0, 1, 2, 3, 1, 89, 0, 3, 2, 1, 1, 5, 0

Offset: 1

Reinhard Zumkeller, Jan 24 2003

a(n) = 0 iff n mod 6 = 0 (A008588);
a(n) = 2 iff n mod 6 = 2 (A016933);
for n>1: a(n)=n iff n is prime (A000040, A008578).

Cf. A080064, A080065, A010872, A010873, A010875, A010877, A010881.

A130910 Sum {0<=k<=n, k mod 16} (Partial sums of A130909).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 175, 186, 198, 211, 225, 240, 240, 241, 243, 246, 250, 255, 261, 268, 276, 285, 295, 306, 318, 331, 345, 360, 360, 361, 363, 366, 370, 375, 381, 388

Offset: 0

Hieronymus Fischer, Jun 11 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Accumulate[Mod[Range[0,60],16]] (* Harvey P. Dale, May 30 2020 *)

a(n)=120*floor(n/16)+A130909(n)*(A130909(n)+1)/2. - G.f.: g(x)=(sum{1<=k<16, k*x^k})/((1-x^16)(1-x)). Also: g(x)=x(15x^16-16x^15+1)/((1-x^16)(1-x)^3).

a(n) = +a(n-1) +a(n-16) -a(n-17). G.f. ( x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +14*x^13 +15*x^14) ) / ( (1+x) *(1+x^2) *(1+x^4) *(1+x^8) *(x-1)^2 ). - R. J. Mathar, Nov 05 2011

A306331 Numbers congruent to 6 or 31 mod 38.

Original entry on oeis.org

6, 31, 44, 69, 82, 107, 120, 145, 158, 183, 196, 221, 234, 259, 272, 297, 310, 335, 348, 373, 386, 411, 424, 449, 462, 487, 500, 525, 538, 563, 576, 601, 614, 639, 652, 677, 690, 715, 728, 753, 766, 791, 804, 829, 842, 867, 880, 905

Offset: 1

Davis Smith, Feb 07 2019

A007310(a(n) + 1) is always a multiple of 19.
A020639(A007310(a(n) + 1)) = 5, 7, 11, 13, 17, or 19.
It equals 5 when n is a term in A273669.
It equals 7 when n is congruent to 3 or 12 (mod 14) but not a term in A273669.
It equals 11 when n is congruent to 4 or 19 (mod 22) but not a case where it equals 5 or 7.
It equals 13 when n is congruent to 5 or 22 (mod 26) (one more than a term in A306285) but not a case where it equals 5, 7, or 11.
It equals 17 when n is congruent to 6 or 29 (mod 34) but not a case where it equals 5, 7, 11, or 13.
For all other cases, it equals 19.
a(n) and (n - 1) have the same remainder (mod 6) (see A010875).

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

Cf. A005408, A005843, A007310, A010875, A020639, A040031, A273669, A306285.

Maple
```
seq(seq(38*i+j, j=[6, 31]), i=0..200);
```

Select[Range[200], MemberQ[{6, 31}, Mod[#, 38]] &]
Union[38Range[30] - 32, 38Range[30] - 7] (* Alonso del Arte, Feb 08 2019 *)

for(n=6, 905, if((n%38==6) || (n%38==31), print1(n, ", ")))

Vec(x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019

(6 to 1108 by 38).union(31 to 1133 by 38).sorted // Alonso del Arte, Feb 08 2019

G.f.: x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
a(n) = 19*n - 10 + 3*(-1)^n. - Wesley Ivan Hurt, Mar 10 2019
a(n) = 19*n - 13 when n is odd and 19*n - 7 when n is even.
a(n) = 19*n - (A040031(n + 1) + 1).
E.g.f.: 7 + (19*x - 10)*exp(x) + 3*exp(-x). - David Lovler, Sep 10 2022

cp's OEIS Frontend

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

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A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

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A204671 a(n) = n^n (mod 6).

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A319707 Filter sequence which records for primes their residue modulo 6, and for all other numbers assigns a unique number.

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A008670 Molien series for Weyl group F_4.

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A010884 Period 5: repeat [1,2,3,4,5].

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A053841 (Sum of digits of n written in base 6) modulo 6.

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