A360911 -id:A360911 - cp's OEIS frontend

A361430 Multiplicative with a(p^e) = e - 1.

1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Vaclav Kotesovec, Mar 11 2023

a(n) is the number of coreful divisors d of n such that n/d is also a coreful divisor of n (a coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n, see A307958). - Amiram Eldar, Aug 15 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10^9 terms)

Cf. A001694, A298826, A307958, A335850 (indices of records).

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A360908 (2*e-1), A360910 (3*e-1), A360911 (3*e-2).

g[p_, e_] := e-1; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 200]

for(n=1, 100, print1(direuler(p=2, n, 1 + 1/(1 - 1/X)^2)[n], ", "))

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,2]-1; f[k,2] = 1); factorback(f); \\ Michel Marcus, Mar 13 2023

from math import prod
from sympy import factorint
def A361430(n): return prod(e-1 for e in factorint(n).values()) # Chai Wah Wu, Apr 15 2025

Dirichlet g.f.: Product_{primes p} (1 + 1/(p^s - 1)^2).
Dirichlet g.f.: zeta(2*s) * zeta(3*s)^2 * Product_{primes p} (1 + 2/p^(4*s) + 2/p^(5*s) - 1/p^(6*s) - 2/p^(7*s) - 2/p^(8*s)).
Let f(s) = Product_{primes p} (1 + 2/p^(4*s) + 2/p^(5*s) - 1/p^(6*s) - 2/p^(7*s) - 2/p^(8*s)), then
Sum_{k=1..n} a(k) ~ f(1/2) * zeta(3/2)^2 * sqrt(n) + zeta(2/3) * (f(1/3) * (log(n) + 6*gamma - 3 + 2*zeta'(2/3)/zeta(2/3)) + f'(1/3)) *  n^(1/3) / 3, where
f(1/2) = Product_{primes p} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 2.20286226691565931157047065666916419062717171359087693723221239...
f(1/3) = Product_{primes p} (1 + 2/p^(4/3) + 2/p^(5/3) - 1/p^2 - 2/p^(7/3) - 2/p^(8/3)) = 6.250573144372477079986352917664218040797528021629950408099536...
f'(1/3) = f(1/3) * Sum_{primes p} (-2*(-8 + p^(1/3) + 4*p^(2/3)) * log(p) / (-2 + p^(2/3) + p - p^(5/3) + p^2)) = -90.898558294301467740374653006294640945295... and gamma is the Euler-Mascheroni constant A001620.
Conjecture: a(n) = abs(A298826(n)).
a(n) > 0 if and only if n is powerful (A001694). - Amiram Eldar, Aug 15 2023

A360910 Multiplicative with a(p^e) = 3*e - 1.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 8, 5, 4, 2, 10, 2, 4, 4, 11, 2, 10, 2, 10, 4, 4, 2, 16, 5, 4, 8, 10, 2, 8, 2, 14, 4, 4, 4, 25, 2, 4, 4, 16, 2, 8, 2, 10, 10, 4, 2, 22, 5, 10, 4, 10, 2, 16, 4, 16, 4, 4, 2, 20, 2, 4, 10, 17, 4, 8, 2, 10, 4, 8, 2, 40, 2, 4, 10, 10, 4, 8, 2

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048785, A226602, A360909, A360911, A360908.

a[n_] := Times @@ ((3*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+2*X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 2/p^(2*s)).

Let f(s) = Product_{primes p} (1 + 2/p^(2*s)), then Sum_{k=1..n} a(k) ~ n*(f(1)*(log(n) + 2*gamma - 1) + f'(1)), where f(1) = Product_{primes p} (1 + 2/p^2) = 2.1908700855532557963501937947188223715671192999357721091330157224657649571..., f'(1) = f(1) * Sum_{primes p} (-4*log(p)/(p^2 + 2)) = -3.559220569509264750413960031425742000438433285978558703470289340806139902... and gamma is the Euler-Mascheroni constant A001620.

A360909 Multiplicative with a(p^e) = 3*e + 2.

Original entry on oeis.org

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s - 1/p^(2*s)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).
Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

Original entry on oeis.org

1, 5, 5, 10, 5, 25, 5, 15, 10, 25, 5, 50, 5, 25, 25, 20, 5, 50, 5, 50, 25, 25, 5, 75, 10, 25, 15, 50, 5, 125, 5, 25, 25, 25, 25, 100, 5, 25, 25, 75, 5, 125, 5, 50, 50, 25, 5, 100, 10, 50, 25, 50, 5, 75, 25, 75, 25, 25, 5, 250, 5, 25, 50, 30, 25, 125, 5, 50, 25, 125, 5, 150

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e).

Cf. A082476.

g[p_, e_] := 5*e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+3*X+X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + 5*p^s/(p^s - 1)^2).

a(n) = A005361(n) * A082476(n).

cp's OEIS Frontend

A361430 Multiplicative with a(p^e) = e - 1.

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A360910 Multiplicative with a(p^e) = 3*e - 1.

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A360909 Multiplicative with a(p^e) = 3*e + 2.

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A360997 Multiplicative with a(p^e) = e + 3.

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A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

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