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.

A226566 Numbers k such that Sum_{d|k} sigma(d)^3/d is an integer, where d are the divisors of k.

Original entry on oeis.org

1, 201, 981, 1962, 3663, 7326, 10791, 12753, 15879, 21582, 25506, 30411, 56898, 60822, 135749, 140283, 172161, 212454, 266727, 280566, 334521, 344322, 360027, 395343, 399267, 407247, 507177, 625878, 669042, 720054, 739674, 790686, 798534, 881919, 1014354, 1221741
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 981 are 1, 3, 9, 109, 327, 981.
sigma(1) = 1, sigma(3) = 4, sigma(9) = 13, sigma(109) = 110, sigma(327) = 440, sigma(981) = 1430.
(1^3/1 + 4^3/3 + 13^3/9 + 110^3/109 + 440^3/327 + 1430^3/981) = 3253822.

Links

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226565

Programs

  • Maple
    with(numtheory); ListA226566:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/a[k],k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226566(10^6);
  • Mathematica
    aQ[n_] := IntegerQ[DivisorSum[n, DivisorSigma[1, #]^3/# &]]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 18 2019 *)
  • PARI
    isok(n) = denominator(sumdiv(n, d, sigma(d)^3/d)) == 1; \\ Michel Marcus, Sep 18 2019

A226563 Numbers k such that sum(d|k, sigma(d)^2) is a multiple of k.

Original entry on oeis.org

1, 2, 10, 185, 370, 3145, 6290, 40885, 53465, 63750, 81770, 106930, 241400, 348750, 427720, 828750, 866200, 1207000, 1306875, 1635449, 2613750, 3138200, 3270898, 7149375, 8054345, 8177245, 14298750, 14725400, 15691000, 16108690, 16354490, 16989375, 30368120
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 3145 are 1, 5, 17, 37, 85, 185, 629, 3145.
sigma(1) = 1, sigma(5) = 6, sigma(17) = 18, sigma(37) = 38, sigma(85) = 108, sigma(185) = 228, sigma(629) = 684, sigma(3145) = 4104.
(1^2 + 6^2 + 18^2 + 38^2 + 108^2 + 228^2 + 684^2 + 4104^2) / 3145 = 5525.

Links

Crossrefs

Cf. A068978, A068986, A226564-A226566

Programs

  • Maple
    with(numtheory); ListA226563:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^2,k=1..nops(a));
if type(b/n,integer) then print(n); fi; od; end: ListA226563(10^6);

Extensions

a(14)-a(33) from Giovanni Resta, Jun 11 2013

A226565 Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.

Original entry on oeis.org

1, 2, 14, 32, 39, 42, 78, 96, 105, 117, 126, 133, 189, 195, 210, 224, 234, 266, 288, 378, 390, 399, 465, 480, 546, 585, 672, 793, 798, 930, 975, 1170, 1197, 1248, 1365, 1470, 1586, 1638, 1862, 1950, 1995, 2016, 2379, 2394, 2646, 2730, 3255, 3360, 3393, 3591
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 189 are 1, 3, 7, 9, 21, 27, 63, 189, sigma(1) = 1, sigma(3) = 4, sigma(7) = 8, sigma(9) = 13, sigma(21) = 32, sigma(27) = 40, sigma(63) = 104, sigma(189) = 320. (1^3 + 4^3 + 8^3 + 13^3 + 32^3 + 40^3 + 104^3 + 320^3) / 189 = 179854.

Links

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226566, A344044.

Programs

  • Maple
    with(numtheory); ListA226565:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/n,k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226565 (10^6);
  • Mathematica
    Select[Range[4000],Divisible[Total[DivisorSigma[1,#]^3&/@Divisors[#]],#]&] (* Harvey P. Dale, Sep 17 2019 *)
s[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Select[Range[3600], Divisible[s[#], #] &] (* Amiram Eldar, Jul 01 2022 *)

A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.

Original entry on oeis.org

1, 11, 19, 71, 41, 209, 71, 367, 226, 451, 155, 1349, 209, 781, 779, 1695, 341, 2486, 419, 2911, 1349, 1705, 599, 6973, 1166, 2299, 2278, 5041, 929, 8569, 1055, 7359, 2945, 3751, 2911, 16046, 1481, 4609, 3971, 15047, 1805, 14839, 1979, 11005, 9266, 6589, 2351, 32205, 3746, 12826, 6479
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Crossrefs

Cf. A060640, A065018, A226564, A344043.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^2/# &]; Array[a, 51] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, sigma(d)^2/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k)^2))
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - p^2*X^2) / ((1 - X) * (1 - p*X)^3 * (1 - p^2*X)))[n], ", ")) \\ Vaclav Kotesovec, May 08 2021

Formula

G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, May 08 2021: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1)^3 * zeta(s-2) / zeta(2*s-2).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * zeta(3) * n^3 / 36. (End)
Showing 1-4 of 4 results.

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