A344299 -id:A344299 - cp's OEIS frontend

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -3, 1, 1, 1, -19, 1, 10, 1, -3, 1, 1, 1, -3, 26, 1, 10, -3, 1, 1, 1, -19, 1, 1, 1, -30, 1, 1, 1, -3, 1, 1, 1, -3, 10, 1, 1, -19, 50, 26, 1, -3, 1, 10, 1, -3, 1, 1, 1, -3, 1, 1, 10, -83, 1, 1, 1, -3, 1, 1, 1, -30, 1, 1, 26, -3, 1, 1, 1, -19

Offset: 1

Ilya Gutkovskiy, May 14 2021

Excess of sum of odd squares dividing n over sum of even squares dividing n.

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

Cf. A002129, A035316, A300853, A321543, A344299.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) # &, IntegerQ[#^(1/2)] &], {n, 1, 80}]
f[p_, e_] := (p^(2*Floor[e/2] + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*Floor[e/2] + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*d)); \\ Michel Marcus, Aug 22 2021

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 2 - (2^(2*floor(f[i,2]/2) + 2) - 1)/3, (f[i,1]^(2*floor(f[i,2]/2) + 2) - 1)/(f[i,1]^2 - 1)));} \\ Amiram Eldar, Nov 15 2022

Multiplicative with a(2^e) = 2 - (2^(2*floor(e/2) + 2) - 1)/3, and a(p^e) = (p^(2*floor(e/2) + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 15 2022

A347176 G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, 4, 1, 1, -1, 1, 1, 1, -5, 1, 4, 1, -1, 1, 1, 1, -1, 6, 1, 4, -1, 1, 1, 1, -5, 1, 1, 1, -4, 1, 1, 1, -1, 1, 1, 1, -1, 4, 1, 1, -5, 8, 6, 1, -1, 1, 4, 1, -1, 1, 1, 1, -1, 1, 1, 4, -13, 1, 1, 1, -1, 1, 1, 1, -4, 1, 1, 6, -1, 1, 1, 1, -5, 13, 1, 1, -1, 1, 1, 1, -1, 1, 4

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

Excess of sum of square roots of odd square divisors of n over sum of square roots of even square divisors of n.

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

Cf. A002129, A002162, A037213, A069290, A344299, A344300.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) #^(1/2) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 3 - 2^(Floor[e/2] + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*sqrtint(d))); \\ Michel Marcus, Aug 22 2021

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 3 - 2^(floor(f[i,2]/2) + 1), (f[i,1]^(floor(f[i,2]/2) + 1) - 1)/(f[i,1] - 1)));} \\ Amiram Eldar, Nov 15 2022

Multiplicative with a(2^e) = 3 - 2^(floor(e/2) + 1), and a(p^e) = (p^(floor(e/2) + 1) - 1)/(p - 1) for p > 2. - Amiram Eldar, Nov 15 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023

A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

The unitary analog of A344299.

The least term that is larger than 2 is a(225) = 4.

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

Cf. A002117, A013661, A016742, A016754, A056624, A344299, A350388, A360157.

f[p_, e_] := If[OddQ[e], 1, 2]; f[2, e_] := If[OddQ[e], 1, 0]; 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]%2, 1, if(f[i, 1] == 2, 0, 2)));}

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1).
Multiplicative with a(2^e) = 1 if e is odd and 0 if e is even, and for p > 2, a(p^e) = 1 if e is odd and 2 if e is even.
Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s - 1)/(4^s + 2^s + 1).
Sum_{k=1..n} a(k) ~ c * n, where c = 5*zeta(2)/(7*zeta(3)) = 0.977451984014... .

cp's OEIS Frontend

A344300 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^(k^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A347176 G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula