A347176 -id:A347176 - cp's OEIS frontend

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

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).

A360165 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares minus the sum of the square roots of the unitary divisors of n that are even squares.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, 4, 1, 1, -1, 1, 1, 1, -3, 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, -3, 8, 6, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 4, -7, 1, 1, 1, -1, 1, 1, 1, 4, 1, 1, 6, -1, 1, 1, 1, -3, 10, 1

Offset: 1

Amiram Eldar, Jan 29 2023

The unitary analog of A347176.

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

Cf. A001620, A306016, A347176, A360162, A360164.

f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := If[OddQ[e], 1, 1 - 2^(e/2)]; 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, if(f[i, 2]%2, 1, 1 - f[i, 1]^(f[i, 2]/2)), 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} (-1)^(d+1)*sqrt(d).
a(n) = A360164(n) - 2 * A360162(n).
Multiplicative with a(2^e) = 1 - 2^(e/2) if e is even and 1 otherwise, 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+2)+2)/(2^(3*s)-2).
Sum_{k=1..n} a(k) ~ (n/Pi^2)*(log(n) + 3*gamma - 1 + 4*log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

cp's OEIS Frontend

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

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