A384047 -id:A384047 - cp's OEIS frontend

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.

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.

A384058 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 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 7, 7, 8, 5, 11, 6, 13, 7, 10, 15, 17, 8, 19, 15, 14, 11, 23, 14, 25, 13, 26, 21, 29, 10, 31, 31, 22, 17, 35, 24, 37, 19, 26, 35, 41, 14, 43, 33, 40, 23, 47, 30, 49, 25, 34, 39, 53, 26, 55, 49, 38, 29, 59, 30, 61, 31, 56, 63, 65, 22, 67, 51, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384042.
Cf. A007310, A065330, A065331, 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), A384057 (3-smooth), this sequence (5-rough).

f[p_, e_] := p^e - If[p < 5, 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] < 5, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if p <= 3, and p^e if p >= 5.
a(n) = n * A047994(n) / A384057(n).
a(n) = A047994(A065331(n)) * A065330(n).
Dirichlet g.f.: zeta(s-1) * ((1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s)) * ((1 - 2/3^s + 1/3^(2*s-1))/(1 - 1/3^s)).
Sum_{k=1..n} a(k) ~ (55/144) * n^2.
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-rough number (i.e., not divisible by any prime smaller than the prime p) is (1/2) * Product_{q prime <= p} (1 - 1/q + 1/(q+1)) * n^2.

A385198 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 prime power (A246655).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 5, 1, 7, 6, 1, 1, 9, 1, 7, 8, 11, 1, 9, 1, 13, 1, 9, 1, 14, 1, 1, 12, 17, 10, 11, 1, 19, 14, 11, 1, 20, 1, 13, 12, 23, 1, 17, 1, 25, 18, 15, 1, 27, 14, 13, 20, 29, 1, 26, 1, 31, 14, 1, 16, 32, 1, 19, 24, 34, 1, 15, 1, 37, 26, 21

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 3 of the values are prime powers, and therefore a(6) = 3.

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

The unitary analog of A116512.
Cf. A047994, A065463, A069513, A077610, A246655, 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), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), A385197 (noncomposite), this sequence (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(Total[1/f @@@ fct])]; Array[a, 100]

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

The unitary convolution of A047994 (the unitary totient phi) with A069513 (the characteristic function of prime powers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A069513(n/d).
a(n) = uphi(n) * Sum_{p^e || n} (1/(p^e-1)), where uphi = A047994, and p^e || n denotes that the prime power p^e unitarily divides n (i.e., p^e divides n but p^(e+1) does not divide n).
a(n) = A385199(n) - A047994(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.26256423811374124133..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = Sum_{p prime}(1/(p^2+p-1)) = 0.37272644617447080939... .

A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 7, 8, 8, 10, 6, 12, 12, 8, 15, 16, 16, 18, 12, 12, 20, 22, 14, 24, 24, 26, 18, 28, 16, 30, 31, 20, 32, 24, 24, 36, 36, 24, 28, 40, 24, 42, 30, 32, 44, 46, 30, 48, 48, 32, 36, 52, 52, 40, 42, 36, 56, 58, 24, 60, 60, 48, 63, 48, 40, 66, 48, 44

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 4 of the values are either 1 or 2, and therefore a(6) = 4.

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

The unitary analog of A126246 (with respect to the definition "the number of integers k from 1 to n such that gcd(n,k) is either 1 or 2").
Cf. A065463, A077610, A138191, 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), A384057 (3-smooth), A384058 (5-rough), this sequence (1 or 2), A385196 (prime), A385197 (noncomposite), A385198 (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; f[2, 1] = 2; 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 && f[i,2] == 1, 0, 1));}

Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and p^e - 1 otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k), the greatest divisor of k that is a unitary divisor of n, is either 1 or a prime power q is a multiplicative function f(n) with f(p^e) = q if p^e = q, and p^e - 1 otherwise.
a(n) = A138191(n) * A047994(n), i.e., a(n) = 2*A047994(n) if n == 2 (mod 4) and A047994(n) otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k) is either 1 or a prime power q is (q/(q-1))*A047994(n) if q is a unitary divisor of n, and A047994(n) otherwise.
Sum_{k=1..n} a(k) ~ (23/40) * 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 ugcd(n, k) is either 1 or a prime p is ((p^4+p^3-1)/(p^4+p^3-p^2)) * c * n^2 / 2, where c = A065463.

A385196 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 prime number.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 0, 5, 1, 3, 1, 7, 6, 0, 1, 8, 1, 3, 8, 11, 1, 7, 0, 13, 0, 3, 1, 14, 1, 0, 12, 17, 10, 0, 1, 19, 14, 7, 1, 20, 1, 3, 8, 23, 1, 15, 0, 24, 18, 3, 1, 26, 14, 7, 20, 29, 1, 18, 1, 31, 8, 0, 16, 32, 1, 3, 24, 34, 1, 0, 1, 37, 24, 3, 16, 38

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 3 of the values are primes, and therefore a(6) = 3.

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

The unitary analog of A117494.
Cf. A010051, A047994, A065463, A077610, 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), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), this sequence (prime), A385197 (noncomposite), A385198 (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; a[1] = 0; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(Total[Boole[# == 1] & /@ fct[[;; , 2]]/(fct[[;; , 1]] - 1)])]; Array[a, 100]

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

from sympy.ntheory import factorint
from sympy import Rational
def a(n: int) -> int:
    if n == 1: return 0
    S, P, F = 0, 1, factorint(n)
    for p, e in F.items():
        P *= p**e - 1
        if e == 1: S += Rational(1, p - 1)
    return int(P * S)
print([a(n) for n in range(1, 79)])  # Peter Luschny, Jun 22 2025

The unitary convolution of A047994 (the unitary totient phi) with A010051 (the characteristic function of prime numbers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A010051(n/d).
a(n) = uphi(n) * Sum_{p || n} (1/(p-1)), where uphi = A047994, and p || n denotes that p unitarily divides n (i.e., the p-adic valuation of n is 1).
a(n) = A385197(n) - A047994(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.21890744964919019488..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = Sum_{p prime}((p^2-1)/(p^2*(p^2+p-1))) = 0.31075288978811405615... .

A385197 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 noncomposite number.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 7, 8, 9, 11, 9, 13, 13, 14, 15, 17, 16, 19, 15, 20, 21, 23, 21, 24, 25, 26, 21, 29, 22, 31, 31, 32, 33, 34, 24, 37, 37, 38, 35, 41, 32, 43, 33, 40, 45, 47, 45, 48, 48, 50, 39, 53, 52, 54, 49, 56, 57, 59, 42, 61, 61, 56, 63, 64, 52, 67, 51

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 5 of the values are noncomposite numbers, and therefore a(6) = 5.

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

The unitary analog of A349338.
Cf. A047994, A065463, A077610, A080339, 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), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), this sequence (noncomposite), A385198 (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(1 + Total[Boole[# == 1] & /@ fct[[;; , 2]]/(fct[[;; , 1]] - 1)])]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2]-1) * (1 + sum(i = 1, #f~, (f[i,2] == 1)/(f[i,1] - 1)));}

The unitary convolution of A047994 (the unitary totient phi) with A080339 (the characteristic function of noncomposite numbers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A080339(n/d).
a(n) = uphi(n) * (1 + Sum_{p || n} (1/(p-1))), where uphi = A047994, and p || n denotes that p unitarily divides n (i.e., the p-adic valuation of n is 1).
a(n) = A385196(n) + A047994(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.92334965064835578762..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = 1 + Sum_{p prime}((p^2-1)/(p^2*(p^2+p-1))) = 1.31075288978811405615... .

A385199 The number of integers k from 1 to n such that the greatest divisor of k that is either 1 or a prime power (A000961).

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 9, 11, 11, 13, 13, 14, 16, 17, 17, 19, 19, 20, 21, 23, 23, 25, 25, 27, 27, 29, 22, 31, 32, 32, 33, 34, 35, 37, 37, 38, 39, 41, 32, 43, 43, 44, 45, 47, 47, 49, 49, 50, 51, 53, 53, 54, 55, 56, 57, 59, 50, 61, 61, 62, 64, 64, 52, 67, 67

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 5 of the values are either 1 or a prime power, and therefore a(6) = 5.

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

The unitary analog of A131233.
Cf. A000961, A010055, A065463, A077610, 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), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), A385197 (noncomposite), A385198 (prime power), this sequence (1 or prime power).

f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct) * (1 + Total[1/f @@@ fct])]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2]-1) * (1 + sum(i = 1, #f~, 1/(f[i,1]^f[i,2] - 1)));}

The unitary convolution of A047994 (the unitary totient phi) with A010055 (the characteristic function of 1 and prime powers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A010055(n/d).
a(n) = uphi(n) * (1 + Sum_{p^e || n} (1/(p^e-1))), where uphi = A047994, and p^e || n denotes that the prime power p^e unitarily divides n (i.e., p^e divides n but p^(e+1) does not divide n).
a(n) = A385198(n) + A047994(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.96700643911290683406......, c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = (1 + Sum_{p prime}(1/(p^2+p-1))) = 1.37272644617447080939... .

A384245 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is an infinitary divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Amiram Eldar, May 23 2025

First differs from A384047 at n = 30.

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 2, 1, 4, 1, 2, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10153 (first 142 rows flattened)

Cf. A050873, A064379, A077609, A384047, A384246 (positions of 1's).

infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* Michael De Vlieger at A077609 *)
T[n_, k_] := Max[Intersection[infdivs[n], Divisors[k]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
infdivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
T(n, k) = vecmax(setintersect(infdivs(n), divisors(k)));

A385200 The sum of the exponents e for the integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a prime power p^e.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 7, 1, 7, 6, 4, 1, 10, 1, 11, 8, 11, 1, 13, 2, 13, 3, 15, 1, 14, 1, 5, 12, 17, 10, 22, 1, 19, 14, 19, 1, 20, 1, 23, 16, 23, 1, 23, 2, 26, 18, 27, 1, 29, 14, 25, 20, 29, 1, 34, 1, 31, 20, 6, 16, 32, 1, 35, 24, 34, 1, 38, 1, 37, 28

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 12 and the integers k from 1 to 12, the greatest divisor of k that is a unitary divisor of 12 are 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1 and 12, respectively. The prime powers are 3 = 3^1, 4 = 2^2, 3 = 3^1, 4 = 2^2 and 3 = 3^1, and the sum of the exponents is a(12) = 1 + 2 + 1 + 2 + 1 = 7.

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

The unitary analog of A122410.

Cf. A047994, A065463, A077610, A246655, A384047, A385198.

a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ (p^e-1) * Total[e/(p^e-1)]]; a[1] = 0; Array[a, 100]

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

a(n) = uphi(n) * Sum_{p^e || n} (e/(p^e-1)), where uphi = A047994, and p^e || n denotes that the prime power p^e unitarily divides n (i.e., p^e divides n but p^(e+1) does not divide n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.31889766457764592387..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = Sum_{p prime}(p^2/((p^2-1)*(p^2+p-1))) = 0.45269528731431531046... .

cp's OEIS Frontend

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.

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A384058 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 5-rough number (A007310).

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A385198 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 prime power (A246655).

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A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

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A385196 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 prime number.

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A385197 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 noncomposite number.

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A385199 The number of integers k from 1 to n such that the greatest divisor of k that is either 1 or a prime power (A000961).

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A384245 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is an infinitary divisor of n.

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