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 1-5 of 5 results.

A348951 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A048272, A056924, A113652, A228441, A333809, A348515, A348952, A348953, A348954, A348955, A348956, A258998.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(n/#) &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348951(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 + x^k).
a(n) = A258998(n) - A348515(n). - Ridouane Oudra, Aug 21 2025

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A000593, A046897, A070039, A109506, A333810, A348608, A348951, A348952, A348954, A348955, A348956, A037213.

Programs

  • Mathematica
    Table[-DivisorSum[n, (-1)^(# + n/#) # &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[k x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348953(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

Formula

G.f.: Sum_{k>=1} k * x^(k*(k + 1)) / (1 + x^k).
a(n) = A037213(n) - A348608(n). - Ridouane Oudra, Aug 21 2025

A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, May 26 2018

Keywords

Links

Crossrefs

Cf. A038548, A048272, A193773 (odd bisection), A348608, A228441, A010052, A348952.

Programs

  • PARI
    {a(n) = polcoeff(sum(k=1, sqrtint(n), x^(k^2)/(1+x^k))+x*O(x^n), n)}
  • PARI
    a(n) = sumdiv(n, d, if (d <= sqrtint(n), (-1)^(d + n/d))); \\ Michel Marcus, Nov 03 2021

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d). - Ilya Gutkovskiy, Nov 02 2021
a(n) = (A228441(n) + A010052(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A348952(n). - Ridouane Oudra, Aug 20 2025

A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

Original entry on oeis.org

0, 1, -1, 1, -1, -1, -1, 3, -1, -1, -1, 6, -1, -1, -4, 3, -1, 2, -1, -1, -4, -1, -1, 10, -1, -1, -4, -1, -1, 7, -1, 7, -4, -1, -6, 2, -1, -1, -4, 12, -1, -4, -1, -1, -9, -1, -1, 16, -1, 4, -4, -1, -1, -4, -6, 14, -4, -1, -1, 13, -1, -1, -11, 7, -6, -4, -1, -1, -4, 11, -1, 8, -1, -1, -9, -1, -8, -4, -1, 20
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A000593, A070039, A109506, A333810, A348660, A348951, A348952, A348953, A348955, A348956.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(n/#) # &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348954(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

Formula

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

A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952, A258998.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(n/# + 1) &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348515(n) = sumdiv(n,d,if((d*d)<=n,(-1)^(1 + (n/d)),0)); \\ Antti Karttunen, Nov 05 2021
  • Python
    from sympy import divisors
def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 + x^k).
a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021
a(n) = A258998(n) - A348951(n). - Ridouane Oudra, Aug 21 2025
Showing 1-5 of 5 results.

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