A344300 -id:A344300 - cp's OEIS frontend

A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, -1, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, -1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, -1, 3, 1, 1, 0, 1, 1, 1, 0, 1, 2

Offset: 1

Ilya Gutkovskiy, May 14 2021

Number of odd squares dividing n minus number of even squares dividing n.

Inverse Moebius transform of A258998.

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

Cf. A046951, A048272, A072691, A258998, A317529, A344300.

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

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

G.f.: Sum_{k>=1} (1 - theta_4(x^k)) / 2.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(2^e) = 1 - floor(e/2), and a(p^e) = 1 + floor(e/2) for p > 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 (A072691). (End)

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

A360159 a(n) is the sum of divisors of n that are odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 50, 26, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 91, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

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

Cf. A016754, A035316, A078434, A344300.

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

a(n) = Sum_{d|n, d odd square} d.
a(n) = (A035316(n) + A344300(n))/2.
Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = (p^(e+2)-1)/(p^2-1) for even e and a(p^e) = (p^(e+1)-1)/(p^2-1) for odd e.
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-4^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/6 = 0.4353958914... .

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... .

A373031 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^k).

Original entry on oeis.org

1, 1, 1, -3, 1, -3, 1, -3, 10, -3, 1, 6, 1, -3, 10, -19, 1, 6, 1, -19, 10, -3, 1, -10, 26, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -46, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 50, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -83, 26, -30, 1, -19, 10, 71

Offset: 1

Ilya Gutkovskiy, May 20 2024

sign

Cf. A038548, A064027, A066839, A095118, A321543, A333781, A333782, A344300, A372625, A373032.

nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d+1) * d^2.

A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, -3, 1, -3, 1, -3, 1, 6, 1, -3, 10, -3, 1, 6, 1, -19, 10, -3, 1, -10, 1, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -10, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 1, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -19, 26, -30, 1, -19, 10, 71

Offset: 1

Ilya Gutkovskiy, May 20 2024

sign

Cf. A056924, A064027, A070039, A321543, A333809, A333810, A339353, A344300, A372625, A373031.

nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(d+1) * d^2.

cp's OEIS Frontend

A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

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

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A360159 a(n) is the sum of divisors of n that are odd squares.

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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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A373031 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^k).

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A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

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