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 1-7 of 7 results.

A366439 The sum of divisors of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 15, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 60, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 72, 48, 72, 54, 120, 72, 120, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 180, 90
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2023

Keywords

Links

Crossrefs

Cf. A000203, A065463, A268335, A366438.
Similar sequences: A062822, A065764, A180114, A362986, A366440.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]
  • PARI
    lista(max) = for(k = 1, max, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(sigma(f), ", ")));
  • Python
    from math import prod
from itertools import count, islice
from sympy import factorint
def A366439_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e&1 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A366439_list = list(islice(A366439_gen(),30)) # Chai Wah Wu, Oct 11 2023

Formula

a(n) = A000203(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/(2*d^2)) * Product_{p prime} (1 + 1/(p^5-p)) = 1.045911669131479732932..., where d = 0.7044422... (A065463) is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = (1/d) * Product_{p prime} (1 + 1/(p^5-p)) = 2 * c * d = 1.4735686365073812503199... .

A366536 The number of unitary divisors of the cubefree numbers (A004709).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Oct 12 2023

Keywords

Comments

The number of unitary divisors of the squarefree numbers (A005117) is the same as the number of divisors of the squarefree numbers (A072048), because all the divisors of a squarefree number are unitary.

Links

Crossrefs

Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.
Similar sequences: A366534, A366538.

Programs

  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]
  • PARI
    lista(max) = for(k = 1, max, my(e = factor(k)[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(2^(#e), ", ")));
  • Python
    from sympy.ntheory.factor_ import udivisor_count
from sympy import mobius, integer_nthroot
def A366536(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_count(m) # Chai Wah Wu, Aug 05 2024

Formula

a(n) = A034444(A004709(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... .

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.

A366442 The sum of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 6, 8, 12, 14, 18, 20, 24, 31, 30, 32, 48, 38, 42, 44, 48, 57, 54, 72, 60, 62, 84, 68, 72, 74, 96, 80, 84, 108, 90, 112, 120, 98, 102, 104, 108, 110, 114, 144, 144, 133, 156, 128, 132, 160, 138, 140, 168, 180, 150, 152, 192, 158, 192, 164, 168, 183, 174, 248
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2023

Keywords

Links

Crossrefs

Cf. A000203, A013661, A007310, A100044, A366441.
Similar sequences: A008438, A062822, A065764, A180114, A362986, A366439, A366440.

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, 2*Floor[3*n/2] - 1]; Array[a, 100]
  • PARI
    a(n) = sigma((3*n)\2 << 1 - 1)
  • Python
    from sympy import divisor_sigma
def A366442(n): return divisor_sigma((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

Formula

a(n) = A000203(A007310(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2) = 1.644934... (A013661).
The asymptotic mean of the abundancy index of the 5-rough numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007310(k) = Pi^2/9 = 1.0966227... (A100044).
In general, the asymptotic mean of the abundancy index of the prime(k)-rough numbers is zeta(2) * Product_{i=1..k-1} (1 - 1/prime(i)^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 1-7 of 7 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.