A365331 -id:A365331 - cp's OEIS frontend

A365332 The sum of divisors of the largest square dividing n.

1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 7, 31, 1, 13, 7, 1, 1, 1, 31, 1, 1, 1, 91, 1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 13, 1, 7, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 31, 7, 1, 1, 1, 31, 121

Offset: 1

Amiram Eldar, Sep 01 2023

All the terms are odd.
The number of these divisors is A365331(n).
The sum of divisors of the square root of the largest square dividing n is A069290(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A005117, A008833, A069290, A078434, A365331.

f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 1 - f[i,2]%2) - 1)/(f[i,1] - 1));}

a(n) = A000203(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(e + 1 - (e mod 2)) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(2*s-2) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 5*zeta(3/2)/Pi^2 = 1.323444812234... .

A375360 The maximum exponent in the prime factorization of the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 3, 3, 3, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Aug 13 2024

Differs from A365331 at n = 1, 36, 72, 100, ... .

Cf. A051903, A356191, A365331, A372601, A375359.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[(If[OddQ[#], #, # + 1]) & /@ e]]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, vecmax(apply(x -> if(x % 2, x, x+1), factor(n)[,2])));

a(n) = A051903(A356191(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum{k>=1} (1 - 1/zeta(2*k)) = 1.98112786070359477197... .

A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 2, 20, 1, 6, 2, 1, 1, 1, 8, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1

Offset: 1

Amiram Eldar, Jan 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290, A005117, A008833, A077591, A365331, A365332, A380163.

Cf. A002117, A013661, A078434.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(2*Floor[e/2]-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1) * f[i, 1]^(2*(f[i, 2]\2)-1)));}

a(n) = A000010(A008833(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(2*floor(e/2)-1) if e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) / (zeta(2*s-1) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3/2)/(zeta(2)*zeta(3)) = 1.32118019580177760682... .

cp's OEIS Frontend

A365332 The sum of divisors of the largest square dividing n.

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A375360 The maximum exponent in the prime factorization of the smallest exponentially odd number that is divisible by n.

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A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

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