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.

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

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Author

Amiram Eldar, May 18 2025

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    f[p_, e_] := p^e - If[e < 2, 1, 0]; 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,2] == 1, 1, 0));}

Formula

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).

A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 10, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 20, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 20, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 40, 48, 20, 66, 32, 44, 24
Offset: 1

Views

Author

Amiram Eldar, May 18 2025

Keywords

Comments

The number of integers k from 1 to n such that the cubefree part (A360539) of gcd(n,k) is 1.

Links

Crossrefs

Cf. A005117, A036966, A078429, A360539, A254926.
The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), this sequence (cubefull), A384041 (exponentially odd), A384042 (5-rough).

Programs

  • Mathematica
    f[p_, e_] := Switch[e, 1, p-1, 2, p^2-p, , (p^3-p^2+1)*p^(e-3)]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, if(f[i,2] == 2, f[i,1]*(f[i,1]-1), (f[i,1]^3-f[i,1]^2+1)*f[i,1]^(f[i,2]-3))));}

Formula

Multiplicative with a(p^e) = (p^3-p^2+1)*p^(e-3) if e >= 3, p*(p-1) if e = 2, and p-1 otherwise.
a(n) >= A384039(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^6) = 0.62159731307414305346... .

A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 13, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 25, 21, 29, 30, 31, 27, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 39, 48, 48, 51, 39, 53, 50, 55, 49, 57, 58, 59, 45, 61, 62, 56, 53, 65, 66, 67, 51
Offset: 1

Views

Author

Amiram Eldar, May 18 2025

Keywords

Links

Crossrefs

Cf. A000010, A268335.
The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), this sequence (exponentially odd), A384042 (5-rough).

Programs

  • Mathematica
    f[p_, e_] := ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1); 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]^2+f[i,1]-1)*f[i,1]^(f[i,2]-1) - (-1)^f[i,2])/(f[i,1] + 1));}

Formula

Multiplicative with a(p^e) = ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1).
a(n) >= A000010(n), with equality if and only if n = 1.
Dirichlet g.f.: (zeta(s-1)*zeta(2*s)/zeta(s)) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/(p^2+1)) = 0.93749428273130025078... .

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46
Offset: 1

Views

Author

Amiram Eldar, May 18 2025

Keywords

Links

Crossrefs

Cf. A000010, A003586, A007310, A065330, A065331, A372671.
The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

Programs

  • Mathematica
    f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Formula

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A384250 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is a powerful number.

Original entry on oeis.org

1, 1, 2, 4, 4, 2, 6, 6, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 12, 25, 12, 21, 24, 28, 8, 30, 18, 20, 16, 24, 36, 36, 18, 24, 24, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 21, 40, 36, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 44
Offset: 1

Views

Author

Amiram Eldar, May 23 2025

Keywords

Links

Crossrefs

Cf. A001694, A006519, A077609.
Analogous sequences: A384039, A384050.
The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is: A384247(1), A384249 (squarefree), this sequence (powerful), A384251 (odd), A384252 (power of 2).

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], p^e, p^e*(1 - 1/p + 1/p^(2^IntegerExponent[e-1, 2]))]; f[p_, 1] := p-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, f[i,1]^f[i,2] * if(!(f[i,2]%2), 1, 1 - 1/f[i,1] + 1/f[i,1]^(1 << valuation(f[i,2]-1, 2)))));}

Formula

Multiplicative with a(p^e) = p^e if e is even, a(p) = p-1, and a(p^e) = p^e * (1 - 1/p + 1/p^A006519(e-1)) if e is odd >= 3.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.7218055778498707651047..., and f(x) = (1 + x - x^2)/(1 + x) + (1 - x) * Sum_{k>=2} x^(2^k + 1) / (1 - x^(2^k));
Showing 1-5 of 5 results.

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