A174961 -id:A174961 - cp's OEIS frontend

A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 4, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 6, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 4, 3, 6, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 8, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 4, 3, 2, 3, 6, 4, 2, 6, 2, 2, 4, 2, 9, 2, 5, 4, 2

Offset: 1

Amiram Eldar, Dec 09 2023

nonn

The terms of A005361 that are larger than 1, since A005361(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A005361, A013929.
Cf. A084936, A174961, A275699, A368038, A368040.
Cf. A059956, A082695, A229099.

Select[Table[Times @@ FactorInteger[n][[;;, 2]], {n, 1, 250}], # > 1 &]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p > 1, print1(p, ", ")));}

a(n) = A005361(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((zeta(2)*zeta(3)/zeta(6)) - 1/zeta(2))/(1-1/zeta(2)) = (A082695 - A059956)/A229099 = 3.406686208821... .

A368038 The sum of non-unitary divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 8, 14, 9, 12, 24, 5, 12, 16, 30, 41, 36, 24, 18, 56, 7, 15, 28, 36, 48, 48, 24, 62, 36, 105, 20, 40, 84, 39, 64, 72, 54, 48, 120, 21, 36, 87, 84, 140, 112, 60, 42, 144, 11, 64, 30, 72, 126, 96, 72, 108, 96, 233, 28, 76, 60, 120, 54, 112, 180, 117, 84

Offset: 1

Amiram Eldar, Dec 09 2023

nonn

The positive terms of A048146, since A048146(k) = 0 if and only if k is squarefree (A005117).

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

Cf. A005117, A048146, A013929.
Cf. A084936, A174961, A275699, A368039, A368040.
Cf. A002117, A013661, A072691, A229099.

f[p_, e_] := (p^(e+1)-1)/(p-1); nusigma[n_] := Module[{fct = FactorInteger[n]}, If[n == 1, 0, Times @@ f @@@ fct - Times @@ (1 + Power @@@ fct)]]; Select[Array[nusigma, 200], # > 0 &]

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(sigma(f) - prod(i=1, #f~, 1+f[i,1]^f[i,2]), ", ")));}

a(n) = A048146(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/2)*(1-1/zeta(3))/(1-1/zeta(2))^2 = 0.899359898779... .

A368040 The powerful part of the nonsquarefree numbers.

Original entry on oeis.org

4, 8, 9, 4, 16, 9, 4, 8, 25, 27, 4, 32, 36, 8, 4, 9, 16, 49, 25, 4, 27, 8, 4, 9, 64, 4, 72, 25, 4, 16, 81, 4, 8, 9, 4, 32, 49, 9, 100, 8, 108, 16, 4, 9, 8, 121, 4, 125, 9, 128, 4, 27, 8, 4, 144, 49, 4, 25, 8, 9, 4, 32, 81, 4, 8, 169, 9, 4, 25, 16, 36, 8, 4, 27

Offset: 1

Amiram Eldar, Dec 09 2023

nonn

The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A013929, A057521.
Cf. A084936, A174961, A275699, A368038, A368039.
Cf. A013661, A229099, A328013.

f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); if(p > 1, print1(p, ", ")));}

a(n) = A057521(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = d/(3*(1-1/zeta(2))^(3/2)) = 4.778771..., and d = A328013.

A368541 The number of exponential divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 4, 4, 2, 4, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 3

Offset: 1

Amiram Eldar, Dec 29 2023

nonn

The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A013929, A049419.
Cf. A084936, A174961, A275699, A368038, A368039, A368040.
Cf. A013661, A059956, A229099, A327837.

f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, numdiv(f[i, 2])); if(p > 1, print1(p, ", ")));}

a(n) = A049419(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (A327837 - A059956)/A229099 = 2.53623753427906735929... .

A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 6, 14, 6, 10, 18, 5, 12, 14, 30, 36, 30, 22, 15, 42, 7, 10, 26, 24, 42, 30, 21, 62, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13

Offset: 1

Amiram Eldar, May 21 2024

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.
The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

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

Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.

Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }

from math import prod, isqrt
from sympy import mobius, factorint
def A373058(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # Chai Wah Wu, Jul 22 2024

a(n) = A336563(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .

cp's OEIS Frontend

A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

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A368038 The sum of non-unitary divisors of the nonsquarefree numbers.

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A368040 The powerful part of the nonsquarefree numbers.

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A368541 The number of exponential divisors of the nonsquarefree numbers.

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A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

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