A358347 -id:A358347 - 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).

A380400 The sum of unitary divisors of n that are perfect powers (A001597).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 9, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 9, 26, 1, 28, 5, 1, 1, 1, 33, 1, 1, 1, 50, 1, 1, 1, 9, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 28, 1, 9, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 18, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Amiram Eldar, Jan 23 2025

First differs from A360720 at n = 72.

The number of unitary divisors of n that are perfect powers is A380398(n).

			a(4) = 5 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2, and 1 + 4 = 5.
a(72) = 18 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2, and 1 + 8 + 9 = 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k)/n^(3/2) and Sum_{k=1..n} A360720(k)/n^(3/2), for n = 1..1000000

Cf. A001597, A005117, A077610, A358347, A360720, A380398.

ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, # &, CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]

a(n) = sumdiv(n, d, d * (gcd(d, n/d) == 1 && (d == 1 || ispower(d))));

a(n) = Sum_{d|n, gcd(d, n/d) == 1} d * [d in A001597], where [] is the Iverson bracket.
a(n) <= A360720(n).
a(n) = 1 if and only if n is squarefree (A005117).

A360161 a(n) is the sum of unitary divisors of n that are odd squares minus the sum of unitary divisors of n that are even squares.

Original entry on oeis.org

1, 1, 1, -3, 1, 1, 1, 1, 10, 1, 1, -3, 1, 1, 1, -15, 1, 10, 1, -3, 1, 1, 1, 1, 26, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, -30, 1, 1, 1, 1, 1, 1, 1, -3, 10, 1, 1, -15, 50, 26, 1, -3, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 10, -63, 1, 1, 1, -3, 1, 1, 1, 10, 1, 1, 26, -3, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

The unitary analog of A344300.

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

Cf. A016742, A016754, A078434, A247041, A344300, A358347, A360160.

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

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1) * d.
a(n) = A360160(n) - 2 * A358347(n).
Multiplicative with a(2^e) = 1 - 2^e if e is even and 1 if e is odd, and for p > 2, a(p^e) = p^e + 1 if e is even and 1 if e is odd.
Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+3)+4)/(2^(3*s)-4).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)*(4*sqrt(2)-1)) = 0.1393911255... .

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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A380400 The sum of unitary divisors of n that are perfect powers (A001597).

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A360161 a(n) is the sum of unitary divisors of n that are odd squares minus the sum of unitary divisors of n that are even squares.

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Mathematica

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