cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Amiram Eldar, Jun 21 2025

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    f[p_, e_] := p^e - 1; f[2, 1] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    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));}

Formula

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

Views

Author

Amiram Eldar, Jun 21 2025

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    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]
  • PARI
    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));}
  • Python
    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

Formula

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

Views

Author

Amiram Eldar, Jun 21 2025

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    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]
  • PARI
    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)));}

Formula

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

Views

Author

Amiram Eldar, Jun 21 2025

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    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]
  • PARI
    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)));}

Formula

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... .

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

Views

Author

Amiram Eldar, Jun 21 2025

Keywords

Examples

			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.

Links

Crossrefs

The unitary analog of A122410.
Cf. A047994, A065463, A077610, A246655, A384047, A385198.

Programs

  • Mathematica
    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]
  • PARI
    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));}

Formula

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... .
Showing 1-5 of 5 results.

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