A010877 -id:A010877 - cp's OEIS frontend

A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

2, 2, 4, 2, 7, 5, 5, 2, 14, 6, 4, 4, 7, 5, 11, 2, 6, 14, 4, 7, 33, 5, 5, 5, 20, 6, 58, 5, 7, 11, 5, 2, 32, 6, 10, 14, 7, 5, 11, 6, 6, 32, 4, 4, 25, 5, 5, 4, 14, 20, 10, 7, 7, 59, 11, 5, 32, 6, 4, 11, 7, 5, 135, 2, 42, 32, 4, 6, 33, 11, 5, 14, 6, 6, 28, 4, 33, 11, 5, 7, 242, 6, 4, 33, 43, 5, 11, 5, 6, 24, 10, 5, 33, 5, 11, 5, 6, 14, 134, 20, 7, 11, 5, 6, 46, 6

Offset: 1

Antti Karttunen, May 09 2017

nonn

This sequence contains, in addition to the information contained in A286364 (which packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). In contrast to A286366, the parity of the highest power of 2 dividing n is not stored.
Thus we have (among other such identities) the following two identities related to equivalence class partitioning:
For all odd i, odd j: a(i) = a(j) <=> A286366(i) = A286366(j).
For all odd i, odd j: a(i) = a(j) => A010877(i) = A010877(j). [On odd numbers the information contained in a(n) is sufficient to determine the value of n modulo 8, one of the 1, 3, 5 or 7.]

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

Cf. A072400, A286364, A286366, A286370, A286371.

from sympy.ntheory.factor_ import digits
from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8
def a(n): return 2*a286364(n) + int(a072400(n)/4) # Indranil Ghosh, May 09 2017

(define (A286369 n) (+ (* 2 (A286364 n)) (floor->exact (/ (A072400 n) 4))))

a(n) = 2*A286364(n) + floor(A072400(n)/4).

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).

A257179 Expansion of (1 + x^5) / ((1 - x) * (1 + x^4)) in powers of x.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 17 2015

			G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + x^11 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A010877, A259042.

a[ n_] := Boole[n != 0] - Boole[Mod[n, 4] == 0] + 2 Boole[Mod[n, 8] == 0];
a[ n_] := -Boole[n == 0] + {1, 1, 1, 0, 1, 1, 1, 2}[[Mod[n, 8, 1]]];
a[ n_] := SeriesCoefficient[ (1 + x^5) / ((1 - x) * (1 + x^4)), {x, 0, Abs@n}];

{a(n) = (n != 0) - (n%4 == 0) + 2*(n%8 == 0)};

{a(n) = -(n==0) + [2, 1, 1, 1, 0, 1, 1, 1][n%8 + 1]};

{a(n) = polcoeff( (1 + x^5) / ((1 - x) * (1 + x^4)) + x * O(x^abs(n)), abs(n))};

Euler transform of length 10 sequence [1, 0, 0, -1, 1, 0, 0, 1, 0, -1].
Moebius transform is length 8 sequence [1, 0, 0, -1, 0, 0, 0, 2].
a(n) is multiplicative with a(2) = 1, a(4) = 0, a(2^e) = 2 if e>2, a(p^e) = 1 if p>2 and a(0) = 1.
G.f.: (1 + x^5) / ((1 - x) * (1 + x^4)).
G.f.: (1 - x^4) * (1 - x^10) / ((1 - x) * (1 - x^5) * (1 - x^8)).
G.f.: -1 + 1 / (1 - x) + 1 / (1 + x^4).
a(n) = a(-n) for all n in Z. a(n+8) = a(n) unless n=0 or n=-8. a(8*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1. a(8*n + 4) = 0.
a(n) = A259042(n+4) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(s)*(1-1/4^s+2/8^s). - Amiram Eldar, Jan 05 2023

More terms from Antti Karttunen, Jul 29 2018

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

A053843 (Sum of digits of n written in base 8) modulo 8.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A010877, A053829, A141803.

Cf. A010060, A053837-A053844.

Table[Mod[Plus @@ IntegerDigits[n, 8], 8], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n) = A010877(A053829(n)). - Andrey Zabolotskiy, May 18 2016

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).

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.

A206546 Period 8: repeat [1, 7, 11, 13, 13, 11, 7, 1].

Original entry on oeis.org

1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 7, 11, 13, 13, 11, 7, 1

Offset: 1

Wolfdieter Lang, Feb 10 2012

For general Modd n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Modd 15 of the positive odd numbers relatively prime to 15 (the positive odd numbers from all reduced residue classes mod 15), shown in A007775. The underlying periodic sequence with period length 30 is [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1], called, with offset 0, P_15 or Modd15.

			Residues Modd 15 of the positive odd numbers relatively prime to 15:
A007775: 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...
Modd 15: 1, 7, 11, 13, 13, 11,  7,  1,  1,  7, 11, 13, 13, 11, ...

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Cf. A007775, A010877.

Cf. A206545 and further crossrefs given there.

PadRight[{},100,{1,7,11,13,13,11,7,1}] (* Harvey P. Dale, Sep 30 2015 *)

a(n)=[1, 7, 11, 13, 13, 11, 7, 1][n%8+1] \\ Charles R Greathouse IV, Jul 17 2016

(define (A206546 n) (list-ref '(1 7 11 13 13 11 7 1) (modulo (- n 1) 8))) ;; Antti Karttunen, Aug 10 2017

a(n) = A007775(n) (Modd 15) := Modd15(A007775(n)), n>=1, with the periodic sequence Modd15 (period length 30) given in the comment section.
O.g.f: x*(1+x^7+7*x*(1+x^5)+11*x^2*(1+x^3)+13*x^3*(1+x))/(1-x^8) = x*(1+x)*(1+6*x+5*x^2+8*x^3+5*x^4+6*x^5+x^6)/(1-x^8).
a(n) = -k^2 + 7k + 1 where k = (n-1) mod 8. - David A. Corneth, Aug 13 2017

A259042 Period 8 sequence [0, 1, 1, 1, 2, 1, 1, 1, ...].

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 17 2015

			G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A010877, A257179.

a[ n_] := {1, 1, 1, 2, 1, 1, 1, 0}[[Mod[n, 8, 1]]];
a[ n_] := SeriesCoefficient[ 1 / (1 - x) - 1 / (1 + x^4), {x, 0, Abs@n}];

{a(n) = 1 + (n%4 == 0) - 2*(n%8 == 0)};

{a(n) = [ 0, 1, 1, 1, 2, 1, 1, 1][n%8 + 1]};

{a(n) = polcoeff( 1 / (1 - x) - 1 / (1 + x^4) + x * O(x^abs(n)), abs(n))};

Euler transform of length 8 sequence [1, 0, 1, -1, 0, -1, 0, 1].
Moebius transform is length 8 sequence [1, 0, 0, 1, 0, 0, 0, -2].
a(n) is multiplicative with a(2) = 1, a(4) = 2, a(2^e) = 0 if e > 2, a(p^e) = 1 if p > 2.
G.f.: x * (1 + x^3)/((1-x)*(1 + x^4)).
G.f.: x * (1 - x^4)*(1 - x^6)/((1-x)*(1 - x^3)*(1 - x^8)).
G.f.: 1/(1-x) - 1/(1 + x^4).
a(n) = a(-n) = a(n+8) for all n in Z.
a(2*n + 1) = a(4*n + 2) = 1.  a(8*n) = 0. a(8*n + 4) = 2.
a(n) = A257179(n+4) unless n = -4.
Dirichlet g.f.: zeta(s) * (1 + 4^(-s) - 2 * 8^(-s)). - Álvar Ibeas, Mar 18 2021

More terms from Antti Karttunen, Jul 29 2018

A278488 a(n) = A276573(n) modulo 8.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028

Cf. A010877, A276573, A276575, A278499.

(define (A278488 n) (modulo (A276573 n) 8))

a(n) = A010877(A276573(n)) = A276573(n) modulo 8.

cp's OEIS Frontend

A286369 Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

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

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A257179 Expansion of (1 + x^5) / ((1 - x) * (1 + x^4)) in powers of x.

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

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A053843 (Sum of digits of n written in base 8) modulo 8.

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

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GAP

Magma

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Sage

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

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