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.

A365171 The number of divisors d of n such that gcd(d, n/d) is a square.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Aug 25 2023

Keywords

Comments

The sum of these divisors is A365172(n).

Links

Crossrefs

Cf. A000005, A000290, A000583, A004524, A005117, A008833, A034444, A046101, A046951, A182448, A365172, A365173.

Programs

  • Mathematica
    f[p_, e_] := Floor[(e + 3)/4] + Floor[(e + 4)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> (x+3)\4 + (x+4)\4, factor(n)[, 2]));
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X)^2 * (1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Formula

Multiplicative with a(p^e) = floor((e + 3)/4) + floor((e + 4)/4) = A004524(e+3).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) == 1 (mod 2) if and only if n is a fourth power (A000583).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s) + 1)).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s) + 1)).
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)) = Pi^2/15 = A182448,
f'(1) = f(1) * Sum_{p prime} 2*log(p) / (p^2 + 1) = f(1) * 0.8852429263675811068149340172820329246145172848406469350087715037483367369...
and gamma is the Euler-Mascheroni constant A001620. (End)

A365174 The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 107, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Aug 25 2023

Keywords

Comments

The number of these divisors is A365173(n).

Links

Crossrefs

Cf. A000203, A033634, A034448, A268335, A365172, A365173.
Cf. A002117, A013661, A013664.

Programs

  • Mathematica
    f[p_, e_] := 1 + p^e + If[EvenQ[e], (p^(e + 1) - p)/(p^2 - 1), (1 + p^(2*Floor[e/4] + 1))*(p^(2*Floor[(e + 1)/4] + 1) - p)/(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]; 1 + p^e + if(e%2, (1 + p^(2*(e\4) + 1))*(p^(2*((e+1)\4) + 1) - p)/(p^2 - 1), (p^(e+1)-p)/(p^2-1)));}

Formula

Multiplicative with 1 + p^e + (p^(e + 1) - p)/(p^2 - 1) if e is even, and 1 + p^e + (1 + p^(2*floor(e/4)+1))*(p^(2*floor((e+1)/4)+1) - p)/(p^2 - 1) if e is odd.
a(n) <= A000203(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034448(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(6)/(2*zeta(3))) * Product_{p prime} (1 + 1/p^3 - 1/p^6) = 0.809912096042... .

A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jan 19 2024

Keywords

Comments

First differs from A366308 at n = 32.
Dirichlet convolution of A295316 with itself.

Links

Crossrefs

Cf. A000005, A005117, A062503, A268335, A295316, A322483, A365173, A366308, A369308.

Programs

  • Mathematica
    f[p_,e_] := If[OddQ[e], 2, e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> if(x%2, 2, x/2), factor(n)[,2]));
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X^2 + X)^2/(1 - X^2)^2)[n], ", ")) \\ Vaclav Kotesovec, Jan 19 2024
  • Python
    from math import prod
from sympy import factorint
def A369307(n): return prod(2 if e&1 else e>>1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Formula

Multiplicative with a(p^e) = 2 is e is odd, and e/2 if e is even.
a(n) >= 1, with equality if and only if n is the square of a squarefree number (A062503).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s)))^2.
From Vaclav Kotesovec, Jan 19 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*s)).
Let f(s) = Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*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 - (2*p^2 + 2*p - 1) / (p^2*(p+1)^2)) = 0.49623881454854881762168565097162197963340069996226074849602334089041678...,
f'(1) = f(1) * Sum_{p prime} 2*(2*p + 1) * log(p) / ((p+1)*(p^2 + p - 1)) = f(1) * 1.49674466685934940187617305887881799198585080518913793200171026177150513...
and gamma is the Euler-Mascheroni constant A001620. (End)

A377708 Numbers k that have a record number of divisors d such that gcd(d, k/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 120, 180, 360, 840, 1260, 2520, 6300, 7560, 12600, 27720, 69300, 83160, 138600, 360360, 900900, 1081080, 1801800, 5405400, 12612600, 18378360, 30630600, 91891800, 214414200, 349188840, 581981400, 1745944200, 4073869800, 8031343320, 12221609400
Offset: 1

Views

Author

Amiram Eldar, Nov 04 2024

Keywords

Comments

First differs from A365681 at n = 22.
First differs from A377140 at n = 95.
Indices of records in A365173.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, ... (see the link for more values).

Links

Crossrefs

Cf. A268335, A365173, A365681, A377140.
Subsequence of A025487.

Programs

  • Mathematica
    f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq
Showing 1-4 of 4 results.

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