A366537 -id:A366537 - cp's OEIS frontend

A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

1, 3, 4, 6, 12, 8, 9, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 36, 42, 28, 30, 72, 32, 33, 48, 54, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 54, 84, 72, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 108, 90, 112

Offset: 1

Amiram Eldar, Oct 12 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034448, A065463, A077610, A268335, A366439, A366534.

Similar sequences: A034676, A366537, A366539.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, OddQ], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 1.92835521961603199612..., d = A065463 is the asymptotic density of the exponentially odd numbers.
The asymptotic mean of the unitary abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = c * d = 1.35841479521454692063... .

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

Amiram Eldar, Oct 12 2023

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.

Similar sequences: A366534, A366538.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

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), ", ")));

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

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

A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 17, 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, 68, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104

Offset: 1

Amiram Eldar, Oct 12 2023

Also the sum 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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034448, A049417, A077609, A077610, A138302, A271727, A366538.

Similar sequences: A034676, A366535, A366537.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A138302(n)).
a(n) = A049417(A138302(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} f(1/p) = 1.58107339851877782285..., d = A271727 is the asymptotic density of A138302, and f(x) = 1 + x^2 + 2 * Sum_{k>=2} (x^(2^k)-x^(2^k+1)).
The asymptotic mean of the unitary abundancy index of A138302: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A138302(k) = c * d = 1.37948208055913856387... .

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

Amiram Eldar, Feb 15 2024

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A002117, A013663.

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 &]

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, ", ")));}

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

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

Amiram Eldar, Apr 02 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A004709, A047994.

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

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]

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)));

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

Amiram Eldar, Apr 02 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A004709, A191414.

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

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]

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)));

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... .

cp's OEIS Frontend

A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

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

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A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

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A369889 The sum of squarefree divisors of the cubefree numbers.

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

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

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