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.

A369717 The sum of divisors of the smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 7, 13, 7, 31, 91, 57, 15, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 195, 31, 1281, 40, 399, 871, 2821, 993, 63, 1729, 2149, 1767, 91, 1407, 2667, 2379, 465, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A000203, A001694, A197863, A369716, A369718, A369720.
Cf. A002117, A013661, A183699.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] == 1, f[i,2] = 2)); sigma(f);}

Formula

a(n) = A000203(A197863(n)).
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
a(n) >= A000203(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .

A369719 The number of divisors of the smallest cubefull number that is a multiple of n.

Original entry on oeis.org

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 5, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 6, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 20, 4, 16, 16, 16, 4, 16, 16, 16, 16, 16, 4, 64, 4, 16, 16, 7, 16, 64, 4, 16, 16, 64, 4, 16, 4
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A000005, A036966, A356193, A369716, A369720, A369721.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, 4, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x <= 2, 4, x+1), factor(n)[, 2]));
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 * ((1 + 2*X - 3*X^2 + X^4)))[n], ", ")) \\ Vaclav Kotesovec, Jan 30 2024

Formula

a(n) = A000005(A356193(n)).
Multiplicative with a(p) = 4 for e <= 2, and a(p^e) = e+1 for e >= 3.
a(n) >= A000005(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^s - 3/p^(2*s) + 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + 1/p^(6*s) - 2/p^(5*s) - 2/p^(4*s) + 8/p^(3*s) - 6/p^(2*s)). - Vaclav Kotesovec, Jan 30 2024

A369721 The sum of unitary divisors of the smallest cubefull number that is a multiple of n.

Original entry on oeis.org

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 17, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A034448, A036966, A356193, A369718, A369719, A369720.
Cf. A002117, A013662, A183700.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, p^3 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= 2, 1 + f[i,1]^3, 1 + f[i,1]^f[i,2]));}

Formula

a(n) = A034448(A356193(n)).
Multiplicative with a(p) = p^3 + 1 for e <= 2, and a(p^e) = p^e + 1 for e >= 3.
a(n) >= A034448(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(s-1) - 1/p^(2*s-4) + 1/p^(4*s-4) - 1/p^(4*s-3) ).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^12 - 2/p^13 + 1/p^14) = 0.65803546696642353777... .

A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 63, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240
Offset: 1

Views

Author

Amiram Eldar, Jan 31 2024

Keywords

Comments

First differs from A369720 at n = 16.

Links

Crossrefs

Cf. A000203, A335988, A356192, A365349, A369720, A369757, A369759.
Cf. A013662, A013664.

Programs

  • Mathematica
    f[p_, e_] := (p^If[OddQ[e], Max[e, 3] + 1, e + 2] - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^if(f[i,2]%2, max(f[i,2], 3) + 1, f[i,2] + 2) - 1)/(f[i,1] - 1));}
  • Python
    from math import prod
from sympy import factorint
def A369758(n): return prod((p**((3 if e==1 else e)+1+(e&1^1))-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Feb 03 2024

Formula

a(n) = A000203(A356192(n)).
Multiplicative with a(p) = p^3 + p^2 + p + 1, a(p^e) = (p^(e+1)-1)/(p-1) for an odd e >= 3, and a(p^e) = (p^(e+2)-1)/(p-1) for an even e.
a(n) >= A000203(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-5) - 1/p^(3*s-4) - 1/p^(3*s-3) + 1/p^(4*s-5) + 1/p^(4*s-4)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(6) * Product_{p prime} (1 - 1/p^4 - 1/p^6 + 1/p^10 + 1/p^11 - 1/p^13) = 1.00040193512214077945... .
Equivalently, c = Product_{p prime} (1 + 1/(p^3*(p^4 - 1)*(p^4 + p^2 + 1))). - Vaclav Kotesovec, Feb 02 2024
Showing 1-4 of 4 results.

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