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.

A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 30, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 60, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 120, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 120, 84, 144
Offset: 1

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Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A000203, A008683, A046100, A307430, A385006.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), this sequence (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Programs

  • Mathematica
    f[p_, e_] := p^(e-3)*(1 + p + p^2 + p^3); f[p_, 1] := 1 + p; f[p_, 2] := 1 + p + p^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^max(e-3,0) * (p^min(e+1,4)-1)/(p-1));}

Formula

a(n) = Sum_{d | n} d * A307430(n/d) = n * Sum_{d | n} A307430(d) / d.
a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3), where mu is the Moebius function (A008683).
Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = p^(e-3) * (1 + p + p^2 + p^3), for e >= 3.
In general, the sum of divisors d of n such that n/d is k-free (not divisible by a k-th power larger than 1) is multiplicative with a(p^e) = p^max(e-k+1,0) * (p^min(e+1,k)-1)/(p-1).
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(4*s).
In general, the sum of divisors d of n such that n/d is k-free has Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(k*s).
Sum_{i=1..n} a(i) ~ (1575 / (2*Pi^6)) * n^2.

A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 13, 12, 18, 12, 24, 14, 24, 24, 26, 18, 36, 20, 36, 32, 36, 24, 52, 30, 42, 37, 48, 30, 72, 32, 53, 48, 54, 48, 72, 38, 60, 56, 78, 42, 96, 44, 72, 72, 72, 48, 104, 56, 90, 72, 84, 54, 111, 72, 104, 80, 90, 60, 144, 62, 96, 96, 106, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A013662, A033634, A268335.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), this sequence (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Programs

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

Formula

Multiplicative with a(p^e) = p^e + (p^(e+1) - 1)/(p^2-1) if e is odd, and p^e + (p^(e+1) - p)/(p^2-1) if e is even.
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.542116283140158741... .

A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 19, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 39, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 57, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 79, 65, 66, 67, 68
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A013661, A036966, A385005.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), this sequence (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Programs

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

Formula

Multiplicative with a(p) = p and a(p^e) = (p^(e+1) - p^e + p^(e-2) - 1)/(p-1) for e >= 2.
Dirichlet g.f.: zeta(s-1) * 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 = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.022486596136980366... .

A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A007310, A013661, A064987, A072044, A072045, A186099.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).

Programs

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

Formula

a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.

A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 29, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 58, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 116, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 116, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A004709, A138302, A353900.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).

Programs

  • Mathematica
    f[p_, e_] := p^e + Sum[p^(e - 2^k), {k, 0, Floor[Log2[e]]}]; 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] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}

Formula

Multiplicative with a(p^e) = p^e + Sum_{k=0..floor(log_2(e))} p^(e-2^k).
a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .
Showing 1-5 of 5 results.

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