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 1-3 of 3 results.

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

Original entry on oeis.org

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 22 2019

Keywords

Links

Crossrefs

Cf. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).
Cf. A001620 (gamma), A002162.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

Formula

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).
a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

Original entry on oeis.org

2, 5, 9, 13, 17, 23, 27, 32, 38, 44, 48, 56, 60, 66, 74, 80, 84, 93, 97, 105, 113, 119, 123, 133, 139, 145, 153, 161, 165, 177, 181, 188, 196, 202, 210, 222, 226, 232, 240, 250, 254, 266, 270, 278, 290, 296, 300, 312, 318, 327, 335, 343, 347, 359, 367, 377, 385, 391, 395, 411, 415, 421, 433, 441, 449, 461, 465, 473, 481
Offset: 1

Views

Author

Antti Karttunen, Oct 12 2015

Keywords

Links

Crossrefs

Cf. A000005, A006218, A099777, A263084, A263085, A060831.
Cf. also A262518, A262519.

Programs

  • Maple
    with(numtheory): seq(add(tau(2*k), k=1..n), n= 1..60); # Ridouane Oudra, Aug 24 2019
  • Mathematica
    Accumulate[DivisorSigma[0, 2 Range@ 69]] (* Michael De Vlieger, Oct 13 2015 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(2*k)); \\ Michel Marcus, Aug 25 2019
  • Python
    from math import isqrt
def A263086(n): return (t:=isqrt(m:=n>>1))**2-((s:=isqrt(n))**2<<1)+((sum(n//k for k in range(1,s+1))<<1)-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(1) = 2; for n > 1, a(n) = A000005(2*n) + a(n-1) [where A000005(k) gives the number of divisors of k].
Other identities. For all n >= 1:
a(n) = A263084(n) + A263085(n).
a(n) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019
From Ridouane Oudra, Aug 24 2019: (Start)
a(n) = Sum_{k=1..n} A000005(2*k)
a(n) = A006218(n) + A060831(n). (End)

A263085 Partial sums of A099774 (A099774(n) = number of divisors of n-th odd number).

Original entry on oeis.org

1, 3, 5, 7, 10, 12, 14, 18, 20, 22, 26, 28, 31, 35, 37, 39, 43, 47, 49, 53, 55, 57, 63, 65, 68, 72, 74, 78, 82, 84, 86, 92, 96, 98, 102, 104, 106, 112, 116, 118, 123, 125, 129, 133, 135, 139, 143, 147, 149, 155, 157, 159, 167, 169, 171, 175, 177, 181, 187, 191, 194, 198
Offset: 1

Views

Author

Antti Karttunen, Oct 12 2015

Keywords

Links

Crossrefs

Cf. A000005, A001620, A006218, A099774, A263084, A263086.

Programs

  • Mathematica
    FoldList[DivisorSigma[0,2*#2-1]+#1&,1,Range[2,62]] (* Ivan N. Ianakiev, Oct 24 2015 *)
Accumulate[Table[DivisorSigma[0, 2*n-1], {n, 1, 100}]] (* Vaclav Kotesovec, Jan 14 2019 *)
  • PARI
    lista(nn) = {s = 0; forstep (n=1, nn, 2, s += numdiv(n); print1(s, ", "););} \\ Michel Marcus, Oct 12 2015

Formula

a(1) = 1; for n > 1, a(n) = A000005(2*n-1) + a(n-1).
a(n) = A263086(n) - A263084(n).
a(n) ~ n * (log(n) + 2*gamma + 3*log(2) - 1)/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024
Showing 1-3 of 3 results.

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