A336649 -id:A336649 - cp's OEIS frontend

A336651 Odd part of n divided by its largest squarefree divisor.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 3, 5, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 30 2020

The name can be parsed either as "Odd part of {n divided by its largest squarefree divisor}" or "Odd part of n, divided by its largest squarefree divisor". Because A000265 and A003557 commute, both interpretations yield equal results.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A000265, A003557, A007947, A336649, A336652.

f[2, e_] := 1; f[p_, e_] := p^(e-1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = p^(e-1).
a(n) = A003557(A000265(n)) = A000265(A003557(n)).
a(n) = A000265(n) / A204455(n).
A000203(a(n)) = A336649(n).
Dirichlet g.f.: (1 - 1/(2^s-1)^2) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Sep 14 2023

A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 12, 5, 11, 3, 13, 7, 15, 1, 17, 12, 19, 5, 21, 11, 23, 3, 30, 13, 39, 7, 29, 15, 31, 1, 33, 17, 35, 12, 37, 19, 39, 5, 41, 21, 43, 11, 60, 23, 47, 3, 56, 30, 51, 13, 53, 39, 55, 7, 57, 29, 59, 15, 61, 31, 84, 1, 65, 33, 67, 17, 69, 35, 71, 12, 73, 37, 90, 19, 77, 39, 79, 5, 120, 41, 83, 21, 85, 43

Offset: 1

Antti Karttunen, Jul 30 2020

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A000593, A057723, A204455, A330596, A336649, A336651.

f[2, e_] := 1; f[p_, e_] := (p^(e+1) - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (p + p^2 + ... + p^e) = sigma(p^e)-1.
a(n) = A057723(A000265(n)).
a(n) = A204455(n) * A336649(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/21) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = (Pi^2/21) * A330596 = 0.3517974711... . - Amiram Eldar, Nov 12 2022

A360163 a(n) is the sum of the square roots of the divisors of n that are odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

First differs from A336649 at n = 27.

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

Cf. A001620, A069290, A336649.

f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 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, 1] == 2, 1, (f[i, 1]^(floor(f[i, 2]/2)+1) - 1)/(f[i, 1] - 1))); }

a(n) = Sum_{d|n, d odd square} sqrt(d).
a(n) = (A069290(n) + A347176(n))/2.
a(n) = A069290(n) if n is not a multiple of 4.
Multiplicative with a(2^e) = 1, and a(p^e) = (p^(floor(e/2)+1)-1)/(p-1) for p > 2.
Dirichlet g.f.: zeta(s)*zeta(2*s-1)*(1-2^(1-2*s)).
Sum_{k=1..n} a(k) ~ (n/4) * (log(n) + 3*gamma - 1 + 2*log(2)), where gamma is Euler's constant (A001620).

A360164 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

First differs from A336649 at n = 27.

The unitary analog of A360163.

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

Cf. A001620, A056624, A069290, A108269, A306016, A336649, A358347, A360163.

f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := 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, 1] == 2, 1, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1))); }

a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} sqrt(d).
a(n) = A360162(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269).
Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = p^(e/2) + 1 if e is even and 1 if e is odd.
Dirichlet g.f.: (zeta(s)*zeta(2*s-1)/zeta(3*s-1))*(2^(3*s)-2^(s+1))/(2^(3*s)-2).
Sum_{k=1..n} a(k) ~ (2*n/Pi^2)*(log(n) + 3*gamma - 1 + log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A383717 Dirichlet g.f.: Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 0, 3, 10, 11, 6, 13, 14, 15, 0, 17, 6, 19, 10, 21, 22, 23, 0, 5, 26, 0, 14, 29, 30, 31, 0, 33, 34, 35, 6, 37, 38, 39, 0, 41, 42, 43, 22, 15, 46, 47, 0, 7, 10, 51, 26, 53, 0, 55, 0, 57, 58, 59, 30, 61, 62, 21, 0, 65, 66, 67, 34, 69, 70, 71, 0, 73

Offset: 1

Vaclav Kotesovec, May 07 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A007947, A056552, A335341, A336649.

f[p_, e_] := If[e < 3, p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2025 *)

for(n=1, 100, print1(direuler(p=2, n, (1 + X*p + X^2*p))[n], ", "))

Sum_{k=1..n} a(k) ~ c * n^2/2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...

Multiplicative with a(p^e) = p is e <= 2, and 0 otherwise. - Amiram Eldar, May 07 2025

cp's OEIS Frontend

A336651 Odd part of n divided by its largest squarefree divisor.

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A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

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A360163 a(n) is the sum of the square roots of the divisors of n that are odd squares.

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A360164 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares.

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A383717 Dirichlet g.f.: Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)).

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