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 1-4 of 4 results.

A377515 The largest divisor of n that is a term in A276078.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 9, 10, 11, 6, 13, 14, 15, 2, 17, 18, 19, 10, 21, 22, 23, 6, 25, 26, 9, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 18, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Oct 30 2024

Keywords

Comments

First differs from A327937 at n = 625 = 5^4: a(625) = 125, while A327937(625) = 625.
The number of these divisors is A377516(n), and their sum is A377517(n).

Links

Crossrefs

Cf. A000720, A276078, A327937, A377516, A377517, A377518.

Programs

  • Mathematica
    f[p_, e_] := p^Min[PrimePi[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~, f[i,1]^min(primepi(f[i,1]), f[i,2]));}

Formula

Multiplicative with a(p^e) = p^min(pi(p), e), where pi(n) = A000720(n).
a(n) = n if and only if n is in A276078.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1) - p^(pi(p)*s) + p^pi(p))/p^((pi(p)+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^pi(p) * (p+1))) = 0.80906238421914194523... .

A377517 The sum of the divisors of n that are terms in A276078.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 13, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 39, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54
Offset: 1

Views

Author

Amiram Eldar, Oct 30 2024

Keywords

Comments

First differs from A046897 at n = 27 = 3^3: a(27) = 13, while A046897(27) = 40.
The number of these divisors is A377516(n), and the largest of them is A377515(n).

Links

Crossrefs

Cf. A000203, A000720, A013661, A046897, A276078, A377515, A377516, A377520.

Programs

  • Mathematica
    f[p_, e_] := (p^(Min[PrimePi[p], e] + 1) - 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~, (f[i,1]^(min(primepi(f[i,1]), f[i,2]) + 1) - 1)/(f[i,1] - 1));}

Formula

a(n) = A000203(A377515(n)).
Multiplicative with a(p^e) = (p^(min(pi(p), e)+1) - 1)/(p - 1), where pi(n) = A000720(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1))/p^((pi(p)+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(pi(p)+1)) = 1.18603586369737251334... .

A377519 The number of divisors of n that are terms in A207481.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 8, 4, 6, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 4, 4, 2, 12, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Oct 30 2024

Keywords

Comments

The sum of these divisors is A377520(n), and the largest of them is A377518(n).

Links

Crossrefs

Cf. A000005, A013661, A207481, A377516, A377518, A377520.

Programs

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

Formula

a(n) = A000005(A377518(n)).
Multiplicative with a(p^e) = min(p, e) + 1.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^((p+1)*s)).

A380086 The number of unitary divisors of n that are terms in A276078.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 2, 2, 2, 4, 4, 1, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 1, 2, 2, 8, 2, 1, 4, 4, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 4, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4
Offset: 1

Views

Author

Amiram Eldar, Jan 11 2025

Keywords

Links

Crossrefs

Cf. A000720, A005117, A034444, A077610, A276078, A325127, A377516, A380085, A380087, A380089.

Programs

  • Mathematica
    f[p_, e_] := If[e <= PrimePi[p], 2, 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] <= primepi(f[i,1]), 2, 1));}

Formula

a(n) = A034444(A380085(n)).
Multiplicative with a(p^e) = 2 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034444(n) if and only if n is in A276079.
a(n) = A034444(n) if and only if n is in A276078.
a(n) = A377516(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((pi(p)+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(pi(p)+1))) = 0.85808348184674088116... .
Showing 1-4 of 4 results.

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