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.

A372379 The largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70
Offset: 1

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Author

Amiram Eldar, Apr 29 2024

Keywords

Comments

First differs from A350390 at n = 32.
The largest term of A036537 dividing n.
The largest divisor of n whose exponents in its prime factorization are all of the form 2^k-1 (A000225).

Links

Crossrefs

Cf. A000225, A000523, A036537, A138302, A162643, A282940, A350390, A353897, A372380, A372381.

Programs

  • Mathematica
    f[p_, e_] := p^(2^Floor[Log2[e + 1]] - 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^exponent(f[i, 2]+1)-1));}
  • Python
    from math import prod
from sympy import factorint
def A372379(n): return prod(p**((1<<(e+1).bit_length()-1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 30 2024

Formula

Multiplicative with a(p^e) = p^(2^floor(log_2(e+1)) - 1).
a(n) = n if and only if n is in A036537.
a(A162643(n)) = A282940(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.7907361848... = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k))-1 for k >= 1, and f(0) = 0.

A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Apr 22 2025

Keywords

Comments

First differs from A061389 at n = 32.
First differs from A322483 at n = 32.
First differs from A372380 at n = 128 (next differences are at n=128*k, n=2187*k, ...).
The number of divisors of n that are both biquadratefree (A046100) and exponentially odd (A268335), i.e., in A336591. - Amiram Eldar, Apr 22 2025

Links

Crossrefs

Cf. A073184, A365498, A383292.
Cf. A322483, A372380, A061389.
Cf. A046100, A268335, A336591.

Programs

  • Mathematica
    f[p_, e_] := If[e < 3, 2, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X + X^3))[n], ", "))

Formula

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^2 * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...,
f'(1) = f(1) * Sum_{p prime} (2*p^2 - 3*p + 4) * log(p) / ((p-1) * (p^3 + p^2 + 1)) = f(1) * 0.85825768698295295413525347933038488513032293516964600096226328323449...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 2 and 3 otherwise. - Amiram Eldar, Apr 22 2025

A372381 The number of divisors of the largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Apr 29 2024

Keywords

Comments

First differs from A286324 at n = 32, and from A331109 at n = 64.
Also, the number of infinitary divisors of the largest divisor of n whose number of divisors is a power of 2.

Links

Crossrefs

Cf. A000005, A004709, A036537, A037445, A372379, A372380.
Cf. A286324, A331109.

Programs

  • Mathematica
    f[p_, e_] := 2^Floor[Log2[e + 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> 2^exponent(x+1), factor(n)[, 2]));
  • Python
    from math import prod
from sympy import factorint
def A372381(n): return prod(1<<(e+1).bit_length()-1 for e in factorint(n).values()) # Chai Wah Wu, Apr 30 2024

Formula

Multiplicative with a(p^e) = 2^floor(log_2(e+1)).
a(n) = A000005(A372379(n)).
a(n) = A037445(A372379(n)).
a(n) = A000005(n) if and only if n is in A036537.
a(n) <= A372380(n), with equality if and only if n is cubefree (A004709).

A382788 The sum of divisors of n that are numbers whose number of divisors is a power of 2.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 11, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 11, 84, 144, 68, 54
Offset: 1

Views

Author

Amiram Eldar, Apr 05 2025

Keywords

Comments

First differs from A033634 at n = 32.
The sum of the terms of A036537 that divide n.
The number of these divisors is A372380(n) and the largest of them is A372379(n).

Links

Crossrefs

Cf. A000523, A033634, A036537, A372379, A353900, A372380.

Programs

  • Mathematica
    f[p_, e_] := Sum[p^(2^k-1), {k, 0, Floor[Log2[e + 1]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, sum(k = 0, exponent(e[i]+1), p[i]^(2^k-1)));

Formula

Multiplicative with a(p^e) = Sum_{k = 0..floor(log_2(e+1))} p^(2^k-1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 1.13143029377358401678... .
Showing 1-4 of 4 results.

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