cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A348956 a(0) = 1; a(n) = Sum_{d|n, d < n} (-1)^(n/d + 1) * a(d - 1).

Original entry on oeis.org

1, 0, -1, 1, -1, 1, 0, 1, -2, 0, 0, 1, 0, 1, -1, -1, -3, 1, 3, 1, 1, 0, -1, 1, -2, 0, -1, -2, -1, 1, 3, 1, -2, 0, 2, 0, 2, 1, -4, 0, -1, 1, 1, 1, 0, -4, 0, 1, -6, 1, 2, -3, 0, 1, 5, 0, 0, 3, 0, 1, 3, 1, -4, -1, -3, 0, 3, 1, 3, -1, -1, 1, 0, 1, -3, -4, -4, 1, 5, 1, -4
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A067856, A167865, A307776, A308077, A335062, A338639, A343371.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^(n/d + 1) a[d - 1], 0], {d, Divisors[n]}]; Table[a[n], {n, 0, 80}]
nmax = 80; A[] = 0; Do[A[x] = 1 - Sum[(-x)^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    A348956(n) = if(!n,1,sumdiv(n,d,if(dA348956(d-1)*(-1)^(1 + (n/d)),0))); \\ Antti Karttunen, Nov 05 2021

Formula

G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x^2) + x^3 * A(x^3) - x^4 * A(x^4) + ...

A347031 a(n) = 1 - Sum_{k=2..n} (-1)^k * a(floor(n/k)).

Original entry on oeis.org

1, 0, 1, 1, 2, -1, 0, 0, 2, -1, 0, 2, 3, 0, 3, 3, 4, -4, -3, -1, 2, -1, 0, 0, 2, -1, 3, 5, 6, -7, -6, -6, -3, -6, -3, 7, 8, 5, 8, 8, 9, -4, -3, -1, 7, 4, 5, 5, 7, -1, 2, 4, 5, -15, -12, -12, -9, -12, -11, 7, 8, 5, 13, 13, 16, 3, 4, 6, 9, -4, -3, -7, -6, -9, -1
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Comments

Partial sums of A308077.

Links

Crossrefs

Cf. A002321, A025523, A308077, A309288, A347030.

Programs

  • Mathematica
    a[n_] := a[n] = 1 - Sum[(-1)^k a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 75}]
nmax = 75; A[] = 0; Do[A[x] = (1/(1 - x)) (x - Sum[(-1)^k (1 - x^k) A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A347031(n):
    if n <= 1:
        return n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j&1)*(1 if j&1 else -1)*A347031(k1)
        j, k1 = j2, n//j2
    return c+(n+1-j&1)*(1 if j&1 else -1) # Chai Wah Wu, Apr 04 2023

Formula

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x - Sum_{k>=2} (-1)^k * (1 - x^k) * A(x^k)).

A335062 a(n) = 1 - Sum_{d|n, d > 1} (-1)^d * a(n/d).

Original entry on oeis.org

1, 0, 2, 0, 2, -2, 2, 0, 4, -2, 2, 0, 2, -2, 6, 0, 2, -8, 2, 0, 6, -2, 2, 0, 4, -2, 8, 0, 2, -14, 2, 0, 6, -2, 6, 4, 2, -2, 6, 0, 2, -14, 2, 0, 16, -2, 2, 0, 4, -8, 6, 0, 2, -24, 6, 0, 6, -2, 2, 8, 2, -2, 16, 0, 6, -14, 2, 0, 6, -14, 2, 0, 2, -2, 16, 0, 6, -14, 2, 0, 16
Offset: 1

Views

Author

Ilya Gutkovskiy, May 21 2020

Keywords

Comments

Inverse Moebius transform of A308077.

Links

Crossrefs

Cf. A048298, A065091 (positions of 2's), A067824, A067856, A308077, A325144, A335283.

Programs

  • Mathematica
    a[n_] := a[n] = 1 - DivisorSum[n, (-1)^# a[n/#] &, # > 1 &]; Table[a[n], {n, 1, 81}]
  • PARI
    lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 - sumdiv(n, d, if (d>1, (-1)^d*va[n/d]));); va;} \\ Michel Marcus, May 22 2020

Formula

G.f. A(x) satisfies: A(x) = x / (1 - x) - Sum_{k>=2} (-1)^k * A(x^k).

A338639 a(0) = 1; for n > 0, a(n) = -Sum_{d|n, d < n} a(d - 1).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, 0, -1, 0, 0, 0, -1, 2, -1, -1, 1, 1, -1, 1, -1, 1, 0, -1, -1, 4, 0, -3, 0, 1, -1, 3, -1, 0, 0, -2, 0, 4, -1, -2, -2, 3, -1, 1, -1, 0, 2, 0, -1, 4, -1, -4, -1, -2, -1, 5, 0, 2, -1, -2, -1, 7, -1, -4, -1, 1, -2, 1, -1, -1, 1, 3, -1, 6, -1, -5, -2, 0, -1, 1, -1, 4
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2020

Keywords

Links

Crossrefs

Cf. A167865, A281487, A308077, A343371.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = -DivisorSum[n, a[# - 1] &, # < n &]; Table[a[n], {n, 0, 80}]
nmax = 80; A[] = 0; Do[A[x] = 1 - Sum[x^k A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 - x^2 * A(x^2) - x^3 * A(x^3) - x^4 * A(x^4) - ...

A343370 a(1) = 1; a(n) = Sum_{d|n, d < n} (-1)^d * a(d).

Original entry on oeis.org

1, -1, -1, -2, -1, -1, -1, -4, 0, -1, -1, -4, -1, -1, 1, -8, -1, -2, -1, -4, 1, -1, -1, -12, 0, -1, 0, -4, -1, -3, -1, -16, 1, -1, 1, -10, -1, -1, 1, -12, -1, -3, -1, -4, 0, -1, -1, -32, 0, -2, 1, -4, -1, -4, 1, -12, 1, -1, -1, -16, -1, -1, 0, -32, 1, -3, -1, -4, 1, -3, -1, -36, -1, -1, 0, -4, 1, -3, -1, -32, 0
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

Links

Crossrefs

Cf. A008683, A053850 (positions of 0's), A056913 (positions of 1's), A067856, A074206, A307778, A308077, A325144.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      add((-1)^d*a(d), d=numtheory[divisors](n) minus {n}))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 70}]
  • PARI
    memoA343370 = Map();
A343370(n) = if(1==n,1,my(v); if(mapisdefined(memoA343370,n,&v), v, v = sumdiv(n,d,if(dA343370(d),0)); mapput(memoA343370,n,v); (v))); \\ Antti Karttunen, Jan 02 2023

Formula

G.f.: x + Sum_{n>=1} (-1)^n * a(n) * x^(2*n) / (1 - x^n).

Extensions

Data section extended up to a(81) by Antti Karttunen, Jan 02 2023

A358276 a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d - 1) * a(d) / d.

Original entry on oeis.org

1, -2, 3, 0, 5, -18, 7, 0, 18, -30, 11, 24, 13, -42, 45, 0, 17, -144, 19, 40, 63, -66, 23, 0, 50, -78, 108, 56, 29, -390, 31, 0, 99, -102, 105, 360, 37, -114, 117, 0, 41, -546, 43, 88, 360, -138, 47, 0, 98, -400, 153, 104, 53, -1080, 165, 0, 171, -174, 59, 1080, 61, -186, 504, 0, 195, -858, 67, 136
Offset: 1

Views

Author

Seiichi Manyama, Mar 30 2023

Keywords

Links

Crossrefs

Cf. A050369, A055615, A308077, A332793.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<2, n, add(b(n/d)*
     (-1)^(d-1), d=numtheory[divisors](n) minus {1}))
    end:
a:= n-> n*b(n):
seq(a(n), n=1..68);  # Alois P. Heinz, Mar 30 2023
  • Mathematica
    a[1] = 1; a[n_] := a[n] = n * DivisorSum[n, (-1)^(n/# - 1) * a[#]/# &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    a(n) = if (n==1, 1, n*sumdiv(n, d, if (dMichel Marcus, Mar 30 2023

Formula

a(n) = n * A308077(n).
If p is prime, a(p) = (-1)^(p-1) * p.
G.f. A(x) satisfies A(x) = x - Sum_{k>=2} (-1)^k * k * A(x^k).

A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -3, 3, -4, 8, -9, 6, -9, 14, -15, 16, -17, 27, -33, 21, -22, 36, -37, 34, -45, 61, -62, 51, -55, 73, -82, 76, -77, 124, -125, 80, -97, 120, -132, 132, -133, 171, -190, 153, -154, 221, -222, 194, -233, 296, -297, 239, -248, 313, -337, 301, -302
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 10 2022

Keywords

Crossrefs

Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1,
      add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..54);  # Alois P. Heinz, Feb 10 2022
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).
G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).

A372626 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1) * a(d).

Original entry on oeis.org

1, -1, 1, 0, 1, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 0, 1, -3, 1, 0, 2, -2, 1, -1, 2, -2, 2, 0, 1, -4, 1, 0, 2, -2, 2, 1, 1, -2, 2, -1, 1, -5, 1, 0, 3, -2, 1, 1, 2, -3, 2, 0, 1, -5, 2, -1, 2, -2, 1, 0, 1, -2, 3, 0, 2, -5, 1, 0, 2, -4, 1, 1, 1, -2, 3, 0, 2, -5, 1, -1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A308077, A337135, A348660.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = DivisorSum[n, (-1)^(n/# + 1) a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 80}]
  • PARI
    a(n) = if (n==1, 1, sumdiv(n, d, if (d^2 <= n, (-1)^(n/d+1)*a(d)))); \\ Michel Marcus, May 09 2024

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * a(k) * x^(k^2) / (1 + x^k).
Showing 1-8 of 8 results.

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