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The sum of non-unitary divisors of the nonsquarefree numbers.
+10
5
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
FORMULA
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... .
MATHEMATICA
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 &]
PROG
(PARI) 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]), ", "))); }
The product of exponents of prime factorization of the nonsquarefree numbers.
+10
5
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
COMMENTS
The terms of A005361 that are larger than 1, since A005361(k) = 1 if and only if k is squarefree ( A005117).
FORMULA
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... .
MATHEMATICA
Select[Table[Times @@ FactorInteger[n][[;; , 2]], {n, 1, 250}], # > 1 &]
PROG
(PARI) lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p > 1, print1(p, ", "))); }
The powerful part of the nonsquarefree numbers.
+10
5
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
COMMENTS
The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree ( A005117).
FORMULA
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.
MATHEMATICA
f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]
PROG
(PARI) 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, ", "))); }
The number of exponential divisors of the nonsquarefree numbers.
+10
2
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
COMMENTS
The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree ( A005117).
FORMULA
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ( A327837 - A059956)/ A229099 = 2.53623753427906735929... .
MATHEMATICA
f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]
PROG
(PARI) 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, ", "))); }
The sum of the aliquot coreful divisors of the nonsquarefree numbers.
+10
1
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
COMMENTS
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).
FORMULA
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = ( A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .
MATHEMATICA
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 &]
PROG
(PARI) 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, ", "))); }
(Python)
from math import prod, isqrt
from sympy import mobius, factorint
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
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