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 1-2 of 2 results.

A318845 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} sigma(j), where sigma(j) = sum of divisors of j (A000203).

Original entry on oeis.org

1, 3, 6, 6, 8, 18, 10, 10, 24, 24, 14, 36, 16, 30, 48, 15, 20, 72, 22, 48, 60, 42, 26, 60, 46, 48, 82, 60, 32, 144, 34, 21, 84, 60, 80, 144, 40, 66, 96, 80, 44, 180, 46, 84, 192, 78, 50, 90, 76, 138, 120, 96, 56, 246, 112, 100, 132, 96, 62, 288, 64, 102, 240, 28, 128, 252, 70, 120, 156, 240
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 04 2018

Keywords

Links

Crossrefs

Cf. A000203, A007429, A007430, A288417, A318768.

Programs

  • Mathematica
    Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[1, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[1, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[1, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3); f[2, e_] := (e+1)*(e+2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)
  • PARI
    a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, (e+1)*(e+2)/2, (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3)));} \\ Amiram Eldar, May 26 2025

Formula

G.f.: Sum_{k>=1} A007429(k)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(A007429(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
From Amiram Eldar, May 26 2025: (Start)
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3) for an odd prime p.
Dirichlet g.f: zeta(s-1) * zeta(s)^2 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^6/864 = 1.112718... . (End)

A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(1/(i*j*k)).

Original entry on oeis.org

1, 1, 3, 15, 69, 477, 4167, 34731, 333225, 4058073, 48535659, 638782119, 9690930477, 146665611765, 2428164153711, 44904494549763, 820664075440593, 16238018609968689, 350155700132388435, 7568774583230565567, 175171222712837235861, 4318996957424273510541, 107317465474650443023383
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Links

Crossrefs

Cf. A000005, A007425, A168243, A280473, A318414, A318696, A318768, A318966.

Programs

  • Maple
    a:=series(mul(mul(mul((1+x^(i*j*k))^(1/(i*j*k)),k=1..55),j=1..55),i=1..55),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[(-1)^(k/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(tau_3(k)/k), where tau_3 = A007425.
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1) * Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.