A360162 -id:A360162 - cp's OEIS frontend

A360164 a(n) is the sum of the square roots of the unitary 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, 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).

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

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

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