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 1-4 of 4 results.

A348952 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

Links

Crossrefs

Cf. A048272, A056924, A228441, A258453, A305152, A333809, A348515, A348951, A348953, A348954, A348955, A348956, A010052.

Programs

  • Mathematica
    Table[-DivisorSum[n, (-1)^(# + n/#) &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A348952(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

Formula

G.f.: Sum_{k>=1} x^(k*(k + 1)) / (1 + x^k).
For p odd prime, a(p) = a(p^2) = -1. - Bernard Schott, Nov 22 2021
a(n) = (A010052(n) - A228441(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A305152(n). - Ridouane Oudra, Aug 20 2025

A348608 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Oct 25 2021

Keywords

Links

Crossrefs

Cf. A000593, A006005, A046897, A066839, A109506, A305152, A333782, A037213, A348953.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(# + n/#) # &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[k x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, if (d<=sqrt(n), (-1)^(d + n/d)*d)); \\ Michel Marcus, Oct 25 2021

Formula

G.f.: Sum_{k>=1} k * x^(k^2) / (1 + x^k).
a(n) = 1 if n = 1 or n is an odd prime (A006005) or n = 4 or n = 8. - Bernard Schott, Dec 18 2021
a(n) = A037213(n) - A348953(n). - Ridouane Oudra, Aug 21 2025

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

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).
Showing 1-4 of 4 results.

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