A382419 -id:A382419 - cp's OEIS frontend

A376366 The number of non-unitary prime divisors of the cubefree numbers.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.2.

Cf. A002117, A004709, A046660, A056170, A088453, A368779, A369427, A376362, A376365, A382419.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], Nothing]]; Array[f, 150]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#select(x -> x == 2, e), ", ")));}

a(n) = A056170(A004709(n)).
a(n) = A369427(A004709(n)).
Sum_{A004709(k) <= x} a(k) = c * x + O(sqrt(x)/log(x)), where c = (1/zeta(3)) * Sum_{p prime} ((p-1)/(p^3-1)) = 0.24833233043359932037... (Das et al., 2025).
a(n) = log_2(A382419(n)). - Amiram Eldar, Mar 25 2025
Sum_{k=1..n} a(k) ~ c * n, where c = Sum_{p prime} ((p-1)/(p^3-1)) = 0.29850959207541746... - Vaclav Kotesovec, Mar 25 2025 (according to the above formula)
From Amiram Eldar, Apr 05 2025: (Start)
a(n) = A046660(A004709(n)).
a(n) = A368779(n) - A376365(n). (End)

A382421 The product of exponents in the prime factorization of the noncubefree numbers.

Original entry on oeis.org

3, 4, 3, 3, 5, 3, 4, 3, 3, 6, 6, 4, 4, 3, 5, 3, 6, 4, 3, 3, 7, 3, 3, 8, 3, 5, 4, 3, 4, 3, 3, 6, 6, 4, 9, 5, 3, 4, 5, 3, 3, 8, 3, 3, 4, 3, 10, 3, 3, 4, 3, 6, 8, 3, 4, 3, 3, 3, 5, 6, 4, 3, 3, 3, 7, 6, 8, 4, 3, 5, 3, 12, 3, 6, 3, 3, 4, 3, 5, 5, 3, 4, 6, 6, 9, 3, 3

Offset: 1

Amiram Eldar, Mar 25 2025

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

Cf. A002117, A013661, A013664, A005361, A046099, A082695, A330594, A368039, A382419.

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; s /@ Select[Range[600], noncubeFreeQ]

list(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecprod(e), ", "))); }

a(n) = A005361(A046099(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)*zeta(3)^2/zeta(6) - zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3))/(zeta(3) - 1) = (A082695 - A330594) * A002117 / (A002117 - 1) = 4.97723390794900554553... .

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

A376366 The number of non-unitary prime divisors of the cubefree numbers.

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A382421 The product of exponents in the prime factorization of the noncubefree 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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