A358039 -id:A358039 - cp's OEIS frontend

A379717 The second Jordan totient function applied to the cubefree numbers.

1, 3, 8, 12, 24, 24, 48, 72, 72, 120, 96, 168, 144, 192, 288, 216, 360, 288, 384, 360, 528, 600, 504, 576, 840, 576, 960, 960, 864, 1152, 864, 1368, 1080, 1344, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 2352, 1800, 2304, 2016, 2808, 2880, 2880, 2520, 3480, 2304

Offset: 1

Amiram Eldar, Dec 31 2024

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

Cf. A002117, A004709, A007434, A013661, A358039 (analogous with J_1 = phi), A379715, A379716, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := Max[FactorInteger[n][[;;, 2]]] < 3; j2 /@ Select[Range[100], cubeFreeQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
iscubefree(n) = if(n == 1, 1, vecmax(factor(n)[, 2]) < 3);
list(lim) = apply(j2, select(iscubefree, vector(lim, i, i)));

a(n) = A007434(A004709(n)).
Sum_{n>=1} 1/a(n) = zeta(2) * zeta(4) / zeta(8) = 35 / (2*Pi^2) = 1.77312071374091100026... .
In general, Sum_{m cubefree} 1/J_k(m) = zeta(k) * zeta(2*k) / zeta(4*k), for k >= 2, where J_k is the k-th Jordan totient function.
In general, Sum_{m k-free} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^(2*k)), for k >= 2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3)^3 * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 1.23061243656940899916... . - Amiram Eldar, Jan 03 2025

A358040 a(n) is the number of divisors of the n-th cubefree number.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 4, 2, 6, 2, 4, 4, 2, 6, 2, 6, 4, 4, 2, 3, 4, 6, 2, 8, 2, 4, 4, 4, 9, 2, 4, 4, 2, 8, 2, 6, 6, 4, 2, 3, 6, 4, 6, 2, 4, 4, 4, 2, 12, 2, 4, 6, 4, 8, 2, 6, 4, 8, 2, 2, 4, 6, 6, 4, 8, 2, 4, 2, 12, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 2, 6, 6, 9, 2

Offset: 1

Amiram Eldar, Oct 29 2022

nonn

The analogous sequence with squarefree numbers is A072048.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.

Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.

DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]

from sympy import mobius, integer_nthroot, divisor_count
def A358040(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 divisor_count(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000005(A004709(n)).

Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).

A374456 The Euler phi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 8, 12, 18, 28, 8, 30, 16, 20, 16, 24, 36, 18, 24, 16, 40, 12, 42, 22, 46, 32, 52, 18, 40, 24, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 40, 88, 72, 60, 46, 72, 32, 96

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A000010, A065463, A268335, A307868.
Similar sequences related to phi: A002618, A049200, A323333, A358039.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374457.

f[p_, e_] := If[OddQ[e], (p-1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A000010(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 / A065463^2 = 0.95051132596733153581... .

A371412 Euler totient function applied to the cubefull numbers (A036966).

Original entry on oeis.org

1, 4, 8, 18, 16, 32, 54, 100, 64, 72, 162, 128, 294, 144, 256, 500, 216, 486, 288, 400, 512, 432, 1210, 576, 648, 800, 1024, 1458, 2028, 2058, 864, 1176, 2500, 1800, 1152, 1296, 1600, 2048, 4624, 2000, 1728, 2352, 1944, 4374, 6498, 2304, 2592, 3200, 4096, 5292, 4000

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A000010, A013661, A036966, A098198.

Similar sequences: A323333, A358039, A371413, A371414.

Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p-1) * p^(e-1)), Nothing]]]; Array[f, 20000]

lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(eulerphi(f), ", ")));}

a(n) = A000010(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*p)) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 3/p^4 + 1/p^5) = 1.65532418864085918623... .

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A379717 The second Jordan totient function applied to the cubefree numbers.

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A358040 a(n) is the number of divisors of the n-th cubefree number.

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A374456 The Euler phi function values of the exponentially odd numbers (A268335).

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A371412 Euler totient function applied to the cubefull numbers (A036966).

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