A367991 -id:A367991 - cp's OEIS frontend

A384554 The sum of the infinitary divisors of n that are cubefree.

1, 3, 4, 5, 6, 12, 8, 7, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 28, 26, 42, 13, 40, 30, 72, 32, 3, 48, 54, 48, 50, 38, 60, 56, 42, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 39, 72, 56, 80, 90, 60, 120, 62, 96, 80, 5, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Jun 03 2025

The number of these divisors is A368883(n), and the largest of them is A384555(n).

The sum of the infinitary divisors of n that are squarefree (A005117) is A367991(n).

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

Cf. A000203, A005117, A004709, A013666, A049417, A073185, A077609, A367991, A368883, A384555.

f[p_, e_] := Switch[Mod[e, 4], 0, 1, 1, p+1, 2, p^2+1, 3, p^2+p+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; [1, p+1, p^2+1, p^2+p+1][e%4+1]);}

from math import prod
from sympy import factorint
def A384554(n): return prod((1,p+1,p**2+1,p*(p+1)+1)[e&3] for p,e in factorint(n).items()) # Chai Wah Wu, Jun 03 2025

Multiplicative with a(p^e) = 1 if e == 0 (mod 4), p + 1 if e == 1 (mod 4), p^2 + 1 if e == 2 (mod 4), and p^2 + p + 1 if e == 3 (mod 4).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) <= A049417(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s + 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-1) + 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(8) * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 2/p^4 - 1/p^5 - 1/p^7) = 1.2351002232125595782019... .

A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000203, A048250, A268335, A336643, A350388, A367991.

Cf. A307868.

f[p_, e_] := If[EvenQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(!(f[i,2]%2), f[i,1]+1, 1));}

Multiplicative with a(p^e) = p + 1 if e is even and 1 otherwise.
a(n) = A048250(A350388(n)).
a(n) = A000203(A336643(n)).
a(n) = A048250(n)/A367991(n).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} ((p^(2*s) - p) * (p^(2*s) + p^s + p) / ((p^s+1) * p^(3*s))).
Let f(s) = Product_{p prime} ((p^(2*s)-p) * (p^(2*s)+p^s+p) / ((p^s+1) * p^(3*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321167...
f'(1) = f(1) * Sum_{p prime} (7*p + 5) * log(p) / ((p-1)*(p+1)*(p+2)) = f(1) * 3.0570993566532132522378281945383016697995408795919384628849894110222383828...
and gamma is the Euler-Mascheroni constant A001620. (End)

A380163 a(n) is the value of the Euler totient function when applied to the squarefree part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 14 2025

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

Cf. A000010, A005117, A007913, A013662, A028982, A055076, A367991, A380162.

f[p_, e_] := If[OddQ[e], p-1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, f[i, 1]-1, 1));}

a(n) = A000010(A007913(n)).
a(n) >= 1, with equality if and only if n is in A028982.
a(n) <= A000010(n), with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p-1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.46350438981962928756...

cp's OEIS Frontend

A384554 The sum of the infinitary divisors of n that are cubefree.

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A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.

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A380163 a(n) is the value of the Euler totient function when applied to the squarefree part of n.

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