A384058 -id:A384058 - cp's OEIS frontend

A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 52, 55, 49, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A063659.
Cf. A000010, A005117, A055231, A057521, A065466, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), this sequence (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 0, 1));}

Multiplicative with a(p) = p and a(p^e) = p^e - 1 if e >= 2.
a(n) = n * A047994(n) / A384050(n).
a(n) = A047994(A057521(n)) * A055231(n) = A000010(A055231(n)) * A057521(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^3*(p+1))) = 0.947733... (A065466).

A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 21, 25, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 36, 37, 38, 39, 35, 41, 42, 43, 44, 45, 46, 47, 45, 49, 50, 51, 52, 53, 52, 55, 49, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A254926.
Cf. A004709, A065468, A360539, A360540, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), this sequence (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 0, 1));}

Multiplicative with a(p^e) = p^e if e <= 2, and p^e - 1 if e >= 3.
a(n) = n * A047994(n) / A384051(n).
a(n) = A047994(A360540(n)) * A360539(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(3*s) + 1/p^(4*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^5*(p+1))) = 0.988504... (A065468).

A384050 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a powerful number.

Original entry on oeis.org

1, 1, 2, 4, 4, 2, 6, 8, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 16, 25, 12, 27, 24, 28, 8, 30, 32, 20, 16, 24, 36, 36, 18, 24, 32, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 27, 40, 48, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384039.
Cf. A000010, A001694, A055231, A057521, A330596, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), this sequence (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 1, 0));}

Multiplicative with a(p) = p-1, and p^e if e >= 2.
a(n) = n * A047994(n) / A384048(n).
a(n) = A047994(A055231(n)) * A057521(n) = A000010(A055231(n)) * A057521(n).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).

A384055 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is odd.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 7, 9, 5, 11, 9, 13, 7, 15, 15, 17, 9, 19, 15, 21, 11, 23, 21, 25, 13, 27, 21, 29, 15, 31, 31, 33, 17, 35, 27, 37, 19, 39, 35, 41, 21, 43, 33, 45, 23, 47, 45, 49, 25, 51, 39, 53, 27, 55, 49, 57, 29, 59, 45, 61, 31, 63, 63, 65, 33, 67, 51, 69

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A026741.
Cf. A000265, A006519, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), this sequence (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[p == 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] == 2, 1, 0));}

Multiplicative with a(2^e) = 2^e-1, and p^e if p is an odd prime.
a(n) = n * A047994(n) / A384056(n).
a(n) = A047994(A006519(n)) * A000265(n).
Dirichlet g.f.: zeta(s-1) * (1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s).
Sum_{k=1..n} a(k) ~ (5/12) * n^2.

A384056 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a power of 2.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 6, 8, 8, 8, 10, 8, 12, 12, 8, 16, 16, 16, 18, 16, 12, 20, 22, 16, 24, 24, 26, 24, 28, 16, 30, 32, 20, 32, 24, 32, 36, 36, 24, 32, 40, 24, 42, 40, 32, 44, 46, 32, 48, 48, 32, 48, 52, 52, 40, 48, 36, 56, 58, 32, 60, 60, 48, 64, 48, 40, 66, 64, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A062570.
Cf. A000079, A000265, A006519, A065463, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), this sequence (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[p == 2, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] == 2, 0, 1));}

Multiplicative with a(2^e) = 2^e, and p^e-1 if p is an odd prime.
a(n) = n * A047994(n) / A384055(n).
a(n) = A047994(A000265(n)) * A006519(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1-1/2^s)/(1-1/2^(s-1)+1/2^(2*s-1))) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ (3/5) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.

A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cubefull number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 16, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 32, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 32, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384040.
Cf. A036966, A360539, A360540, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), this sequence (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 3, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if e <= 2, and p^e if e >= 3.
a(n) = n * A047994(n) / A384049(n).
a(n) = A047994(A360539(n)) * A360540(n).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.714093594477970831206... .

A384052 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a square.

Original entry on oeis.org

1, 1, 2, 4, 4, 2, 6, 7, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 14, 25, 12, 26, 24, 28, 8, 30, 31, 20, 16, 24, 36, 36, 18, 24, 28, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 26, 40, 42, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A206369.
Cf. A013662, A350388, A350389, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), this sequence (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[OddQ[e], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%2, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if e is odd, and p^e if e is even.
a(n) = n * A047994(n) / A384054(n).
a(n) = A047994(A350389(n)) * A350388(n).
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.74061963657217328604... .

A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A078429.
Cf. A013664, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), this sequence (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[Divisible[e, 3], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%3, 1, 0));}

Multiplicative with a(p^e) = p^e if e is a multiple of 3, and p^e-1 otherwise.
Dirichlet g.f.: zeta(s-1) * zeta(3*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) - 1/p^(3*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.71190515701599590826... .

A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 8, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 24, 24, 26, 27, 21, 29, 30, 31, 32, 33, 34, 35, 24, 37, 38, 39, 40, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 54, 55, 56, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384041.
Cf. A013662, A268335, A350388, A350389, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), this sequence (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%2, 0, 1));}

from math import prod
from sympy import factorint
def A384054(n): return prod(p**e-(e&1^1) for p,e in factorint(n).items()) # Chai Wah Wu, May 21 2025

Multiplicative with a(p^e) = p^e if e is odd, and p^e-1 if e is even.
a(n) = n * A047994(n) / A384052(n).
a(n) = A047994(A350388(n)) * A350389(n).
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881... .

A384057 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 3-smooth number.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 12, 12, 16, 16, 18, 18, 16, 18, 20, 22, 24, 24, 24, 27, 24, 28, 24, 30, 32, 30, 32, 24, 36, 36, 36, 36, 32, 40, 36, 42, 40, 36, 44, 46, 48, 48, 48, 48, 48, 52, 54, 40, 48, 54, 56, 58, 48, 60, 60, 54, 64, 48, 60, 66, 64

Offset: 1

Amiram Eldar, May 18 2025

First differs A372671 from at n = 25.

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

Unitary analog of A372671.
Cf. A003586, A065330, A065331, A065463, A372671, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), this sequence (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[p < 5, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] < 5, 0, 1));}

Multiplicative with a(p^e) = p^e if p <= 3, and p^e-1 if p >= 5.
a(n) = n * A047994(n) / A384058(n).
a(n) = A047994(A065330(n)) * A065331(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1-1/2^s)/(1-1/2^(s-1)+1/2^(2*s-1))) * ((1-1/3^s)/(1-2/3^s+1/3^(2*s-1))) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ (36/55) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.
In general, the average order of the number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a p-smooth number (i.e., not divisible by any prime larger than the prime p) is (1/2) * Product_{q prime <= p} (1 + 1/(q^2+q-1)) * A065463 * n^2.

cp's OEIS Frontend

A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

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A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

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A384050 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a powerful number.

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A384055 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is odd.

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A384056 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a power of 2.

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A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cubefull number.

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A384052 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a square.

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A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

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