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.

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... .

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... .

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)).

A380090 The sum of the unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96
Offset: 1

Views

Author

Amiram Eldar, Jan 12 2025

Keywords

Comments

First differs from A371242 at n = 27.

Links

Crossrefs

Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

Programs

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

Formula

a(n) = A034448(A380088(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034448(n) if and only if n is in A185359.
a(n) = A034448(n) if and only if n is in A207481.
a(n) = A377520(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.