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 1-6 of 6 results.

A366534 The number of unitary divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 2, 8, 2, 4, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 8, 4
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2023

Keywords

Links

Crossrefs

Cf. A034444, A077610, A268335, A366438, A366535.
Similar sequences: A366536, A366538.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]
  • PARI
    lista(max) = for(k = 1, max, my(e = factor(k)[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(2^(#e), ", ")));

Formula

a(n) = A034444(A268335(n)).

A366537 The sum of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104, 100, 96
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2023

Keywords

Links

Crossrefs

Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.
Similar sequences: A034676, A366535, A366539.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # < 3 &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]
  • PARI
    lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));
  • Python
    from sympy.ntheory.factor_ import udivisor_sigma
from sympy import mobius, integer_nthroot
def A366537(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return udivisor_sigma(m) # Chai Wah Wu, Aug 05 2024

Formula

a(n) = A034448(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 1/p^4 - 1/p^5) = 1.665430860774244601005... .
The asymptotic mean of the unitary abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = c / zeta(3) = 1.38548421160152785073... .

A366538 The number of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4
Offset: 1

Views

Author

Amiram Eldar, Oct 12 2023

Keywords

Comments

Also, the number of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

Links

Crossrefs

Cf. A034444, A037445, A077609, A077610, A138302, A366539.
Similar sequences: A366534, A366536.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]
  • PARI
    lista(max) = for(k = 1, max, my(e = factor(k)[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(2^#e, ", ")));

Formula

a(n) = A034444(A138302(n)).
a(n) = A037445(A138302(n)).

A369889 The sum of squarefree divisors of the cubefree numbers.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 4, 18, 12, 12, 14, 24, 24, 18, 12, 20, 18, 32, 36, 24, 6, 42, 24, 30, 72, 32, 48, 54, 48, 12, 38, 60, 56, 42, 96, 44, 36, 24, 72, 48, 8, 18, 72, 42, 54, 72, 80, 90, 60, 72, 62, 96, 32, 84, 144, 68, 54, 96, 144, 72, 74, 114, 24, 60, 96, 168
Offset: 1

Views

Author

Amiram Eldar, Feb 15 2024

Keywords

Comments

The number of squarefree divisors of the n-th cubefree number is A366536(n).

Links

Crossrefs

Cf. A004709, A048250, A062822, A366440, A366536, A366537.
Cf. A002117, A013663.

Programs

  • Mathematica
    f[p_, e_] := p + 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; cubefreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[100], cubefreeQ]
(* or *)
f[p_, e_] := If[e > 2, 0, p + 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]
  • PARI
    lista(kmax) = {my(f, s, p, e); for(k = 1, kmax, f = factor(k); s = prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e < 3, p + 1, 0)); if(s > 0, print1(s, ", ")));}
  • Python
    from math import prod
from sympy import mobius, integer_nthroot, primefactors
def A369889(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p+1 for p in primefactors(m)) # Chai Wah Wu, Aug 12 2024

Formula

a(n) = A048250(A004709(n)).
Sum_{j=1..n} a(j) ~ c * n^2, where c = zeta(3)^2/(2*zeta(5)) = 0.6967413068... .
In general, the formula holds for the sum of squarefree divisors of the k-free numbers with c = zeta(k)^2/(2*zeta(2*k-1))..., for k >= 2.

A382662 The unitary totient function applied to the cubefree numbers (A004709).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 4, 10, 6, 12, 6, 8, 16, 8, 18, 12, 12, 10, 22, 24, 12, 18, 28, 8, 30, 20, 16, 24, 24, 36, 18, 24, 40, 12, 42, 30, 32, 22, 46, 48, 24, 32, 36, 52, 40, 36, 28, 58, 24, 60, 30, 48, 48, 20, 66, 48, 44, 24, 70, 72, 36, 48, 54, 60, 24, 78, 40
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A002117, A004709, A047994.
Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382663.

Programs

  • Mathematica
    f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uphi /@ Select[Range[100], cubeFreeQ]
  • PARI
    uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1); }
iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = apply(uphi, select(iscubefree, vector(lim, i, i)));

Formula

a(n) = A047994(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)^2/2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.41625329674394407438... .

A382663 The unitary Jordan totient function applied to the cubefree numbers (A004709).

Original entry on oeis.org

1, 3, 8, 15, 24, 24, 48, 80, 72, 120, 120, 168, 144, 192, 288, 240, 360, 360, 384, 360, 528, 624, 504, 720, 840, 576, 960, 960, 864, 1152, 1200, 1368, 1080, 1344, 1680, 1152, 1848, 1800, 1920, 1584, 2208, 2400, 1872, 2304, 2520, 2808, 2880, 2880, 2520, 3480, 2880
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2025

Keywords

Links

Crossrefs

Cf. A002117, A004709, A191414.
Similar sequences: A366440, A366536, A366537, A368712, A368779, A369889, A376365, A376366, A379717, A382419, A382662.

Programs

  • Mathematica
    f[p_, e_] := p^(2*e)-1; uj2[1] = 1; uj2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; uj2 /@ Select[Range[100], cubeFreeQ]
  • PARI
    uj2(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = apply(uj2, select(iscubefree, vector(lim, i, i)));

Formula

a(n) = A191414(A004709(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)^3/3) * Product_{p prime} (1 - 2/p^3 + 1/p^4 - 1/p^6 + 1/p^7) = 0.42656661743049439763... .
Showing 1-6 of 6 results.

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