A117659 -id:A117659 - cp's OEIS frontend

A163804 Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.

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

Offset: 0

Michael Somos, Aug 04 2009

			1 - x + x^2 - x^4 + x^5 - x^7 + x^8 - x^10 + x^11 - x^13 + x^14 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1).

A106510(n) = -a(n) unless n=0. Convolution inverse of A117659.

Cf. A102283.

1, seq(2*sin(4*Pi*n/3)/sqrt(3), n=1..100); # Ridouane Oudra, Jan 09 2025

Join[{1},LinearRecurrence[{-1, -1},{-1, 1},105]] (* Ray Chandler, Sep 15 2015 *)

PARI
```
{a(n) = (n==0) + [0, -1, 1][n%3 + 1]}
```
PARI
```
{a(n) = (n==0) - kronecker(-3, n)}
```

Euler transform of length 4 sequence [ -1, 1, 1, -1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 - v - u * (4 - 2*v - u).
a(3*n) = 0 unless n=0, a(3*n + 1) = -1, a(3*n + 2) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+3) = a(n) unless n=0 or n=-3.
G.f.: (1 + x^2) / (1 + x + x^2).
G.f.: A(x) = 1 / (1 + x / (1 + x^2)) = 1 - x / (1 + x / (1 - x / (1 + x))). - Michael Somos, Jan 03 2013
a(n) = A057078(n-2), n>1. - R. J. Mathar, Aug 06 2009
From Ridouane Oudra, Jan 09 2025: (Start)
a(n) = 3*floor((n+1)/3) - n, for n>0.
a(n) = 2*sin(4*Pi*n/3)/sqrt(3), for n>0.
a(n) = - A102283(n), for n>0.
a(n) = - A106510(n), for n>0. (End)

A117656 Number of solutions to x^(k+2)=x^2 mod n for some k>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 6, 9, 10, 11, 12, 13, 14, 15, 12, 17, 18, 19, 20, 21, 22, 23, 18, 25, 26, 21, 28, 29, 30, 31, 20, 33, 34, 35, 36, 37, 38, 39, 30, 41, 42, 43, 44, 45, 46, 47, 36, 49, 50, 51, 52, 53, 42, 55, 42, 57, 58, 59, 60, 61, 62, 63, 40, 65, 66, 67, 68, 69, 70, 71, 54

Offset: 1

Steven Finch, Apr 11 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A117657, A117659.

a[n_] := Product[{p, e} = pe; p^Quotient[e, 2] + (p-1)*p^(e-1), {pe, FactorInteger[n]}]; Array[a, 72] (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); p^(e\2) + (p-1)*p^(e-1))} \\ Andrew Howroyd, Jul 17 2018

Multiplicative with a(p^e) = p^floor(e/2) + (p-1)*p^(e-1) for prime p. - Andrew Howroyd, Jul 17 2018

Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 - 1/(p^2*(p^2 + p + 1))) = 0.47717662698737204270... - Amiram Eldar, Sep 08 2020

A117658 Number of solutions to x^(k+1) = x^k mod n for some k >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 2, 6, 2, 4, 4, 9, 2, 8, 2, 6, 4, 4, 2, 10, 6, 4, 10, 6, 2, 8, 2, 17, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 6, 8, 4, 2, 18, 8, 12, 4, 6, 2, 20, 4, 10, 4, 4, 2, 12, 2, 4, 8, 33, 4, 8, 2, 6, 4, 8, 2, 20, 2, 4, 12

Offset: 1

Steven Finch, Apr 11 2006

If n is prime, then the solutions are x = 0, 1, and so a(n) = 2. - Michael B. Porter, Jul 08 2016

The set of solutions is independent of the choice of k. - Michael B. Porter, Jul 08 2016

			For n = 10, using k = 1, the solutions are x = 0, 1, 5, and 6, so a(10) = 4. - _Michael B. Porter_, Jul 08 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A117659.

f:= proc(n) local F,f;
    F:= ifactors(n)[2];
    mul(1 + f[1]^(f[2]-1), f = F)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 06 2016

a[n_] := Module[{F, f}, F = FactorInteger[n]; Product[1 + f[[1]]^(f[[2]] - 1), {f, F}]]; a[1] = 1; Array[a, 100] (* Jean-François Alcover, Nov 05 2016, after Robert Israel *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Sep 21 2023

a(n) = Sum_{k=1..n} floor((k^n-k^(n-1))/n)-floor((k^n-k^(n-1)-1)/n). - Anthony Browne, Jul 06 2016
Multiplicative with a(p^e) = 1 + p^(e-1) for primes p. - Robert Israel, Jul 06 2016
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s - 1/p^(2*s)). - Amiram Eldar, Sep 21 2023

cp's OEIS Frontend

A163804 Expansion of (1 - x) * (1 - x^4) / ((1 - x^2) * (1 - x^3)) in powers of x.

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A117656 Number of solutions to x^(k+2)=x^2 mod n for some k>=1.

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Mathematica

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A117658 Number of solutions to x^(k+1) = x^k mod n for some k >= 1.

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Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

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