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.

A368331 The number of divisors of the largest term of A054743 that divides of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

First differs from A366145 at n = 27.

Links

Crossrefs

Cf. A000005, A054743, A207481, A368328, A368329, A368330, A368335, A368336.
Cf. A366145.

Programs

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

Formula

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e+1 if e > p.
a(n) = A000005(A368329(n)).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= A000005(n), with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^((p+2)*s-1) + 1/p^((p+1)*s) + 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368333 The largest term of A054744 that divide n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

The largest divisor d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

Links

Crossrefs

Cf. A034444, A048103, A054744, A327939, A368329, A368332, A368334, A368335.

Programs

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

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = p^e if e >= p.
A034444(a(n)) = A368334(n).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= n, with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^(p*s) + 1/p^(p*(s-1)) + 1/p^((p+1)*s-1) - 1/p^((p+1)*(s-1)+1)).

A368330 The number of terms of A054743 that are unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

First differ from A043281 at n = 49.

Links

Crossrefs

Cf. A034444, A054743, A207481, A368328, A368329, A368331, A368334.
Cf. A043281.

Programs

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

Formula

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = 2 if e > p.
a(n) = A034444(A368329(n)).
a(n) >= 1, with equality if and only if n is in A207481.
a(n) <= A034444(n), with equality if and only if n is in A054743.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^((p+1)*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^(p+1)) = 1.13896197534988330925... .

A377518 The largest divisor of n that is a term in A207481.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 54, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69
Offset: 1

Views

Author

Amiram Eldar, Oct 30 2024

Keywords

Comments

The number of these divisors is A377519(n), and their sum is A377520(n).

Links

Crossrefs

Cf. A207481, A368329, A377515, A377519, A377520.

Programs

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

Formula

Multiplicative with a(p^e) = p^min(p, e).
a(n) = n if and only if n is in A207481.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1) - p^(p*s) + p^p)/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^p * (p+1))) = 0.908130438292447963703... .

A368328 The number of terms of A054743 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

The number of divisors d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
The largest of these divisors is A368329(n).

Links

Crossrefs

Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.

Programs

  • Mathematica
    f[p_, e_] := If[e <= p, 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~, if(f[i,2] <= f[i,1], 1, f[i,2] - f[i,1] + 1));}

Formula

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e - p + 1 if e > p.
a(n) >= 1, with equality if and only if n is in A207481.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^((p+1)*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .
Showing 1-5 of 5 results.

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