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.

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A373040 a(n) = (A084218(n) - 1)/12.

Original entry on oeis.org

0, 1, 6, 17, 50, 79, 196, 273, 492, 651, 1210, 1247, 2366, 2549, 3656, 4369, 6936, 6397, 10830, 10267, 14314, 15731, 23276, 19935, 31300, 30759, 39858, 40197, 58870, 47529, 76880, 69905, 88336, 90169, 117846, 100877, 156066, 140791, 172724, 164123, 235340, 186083
Offset: 1

Views

Author

Hugo Pfoertner, May 20 2024

Keywords

Links

Crossrefs

Cf. A000290, A001157, A001159, A084218, A372966, A373039.
Cf. A002117, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 12; Array[a, 35] (* Amiram Eldar, Jan 03 2025 *)
  • PARI
    a(n) = (sigma(n^2, 4)/sigma(n^2, 2) - 1)/12

Formula

From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: (zeta(s-4)/zeta(s-2) - zeta(s))/12.
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(60*zeta(3)) = 0.0143771... . (End)

A055698 Numbers n such that n | (sigma_4(n) + phi(n)^4).

Original entry on oeis.org

1, 2, 6, 13588, 48238, 54490, 69004, 194460, 353228, 577980, 638652, 1478962, 1882188, 2190515, 2677740, 3404598, 3875508, 8456460, 9978863, 16320458, 41199780, 45112860, 76132715, 236405988, 357846865, 682983756, 689274612, 733141332, 1185444052, 1752193128
Offset: 1

Views

Author

Robert G. Wilson v, Jun 09 2000

Keywords

Comments

Sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

Crossrefs

Cf. A001159.

Programs

  • Mathematica
    Do[If[Mod[DivisorSigma[4, n] + EulerPhi[n]^4, n]==0, Print[n]], {n, 1, 10^6}]
  • PARI
    isok(k) = (sigma(k, 4) + eulerphi(k)^4)%k == 0; \\ Jinyuan Wang, Mar 17 2020

Extensions

More terms from Robert G. Wilson v, Mar 11 2013
More terms from Jinyuan Wang, Mar 17 2020

A319757 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(2*k+1)/6).

Original entry on oeis.org

1, -1, -5, -9, -6, 35, 125, 275, 291, -241, -2111, -5989, -10990, -11660, 6454, 68298, 201859, 400794, 546122, 269907, -1175825, -4890783, -11746437, -20668698, -25146121, -7959643, 63707489, 236244458, 546634684, 956731805, 1220119643, 676723572, -1964409479, -8645307595
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

Crossrefs

Cf. A000330, A001157, A001158, A001159, A279215, A281156, A283263, A292386, A292387.

Programs

  • Maple
    a:=series(mul((1-x^k)^(k*(k+1)*(2*k+1)/6),k=1..100),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[(1 - x^k)^(k (2 k + 1) (k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[-Sum[x^k (1 + x^k)/(k (1 - x^k)^4), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d^2 (d + 1) (2 d + 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000330(k).
G.f.: exp(-Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^4)).
G.f.: exp(-Sum_{k>=1} (2*sigma_4(k) + 3*sigma_3(k) + sigma_2(k))*x^k/(6*k)).
Previous Showing 71-73 of 73 results.

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