A069290 -id:A069290 - cp's OEIS frontend

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

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

A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 29 2023

The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).

The unitary analog of A069290.

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

Cf. A001620, A056624, A069290, A306016, A358347.

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

a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).
Multiplicative with 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).
Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A365332 The sum of divisors of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 7, 31, 1, 13, 7, 1, 1, 1, 31, 1, 1, 1, 91, 1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 13, 1, 7, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 31, 7, 1, 1, 1, 31, 121

Offset: 1

Amiram Eldar, Sep 01 2023

All the terms are odd.
The number of these divisors is A365331(n).
The sum of divisors of the square root of the largest square dividing n is A069290(n).

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

Cf. A000203, A005117, A008833, A069290, A078434, A365331.

f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 1 - f[i,2]%2) - 1)/(f[i,1] - 1));}

a(n) = A000203(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(e + 1 - (e mod 2)) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(2*s-2) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 5*zeta(3/2)/Pi^2 = 1.323444812234... .

A380325 The sum of the square roots of the squares that divide the n-th exponentially odd number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Jan 20 2025

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

Cf. A013661, A065463, A069290, A268335, A330596, A380323, A380324.

f[p_, e_] := If[OddQ[e], (p^((e+1)/2) - 1)/(p - 1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]^((f[i, 2]+1)/2) - 1)/(f[i, 1] - 1), 0));}
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

a(n) = A069290(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * A330596 / d = 1.74789521005721521109..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A345320 Sum of the divisors of n whose square does not divide n.

Original entry on oeis.org

0, 2, 3, 4, 5, 11, 7, 12, 9, 17, 11, 25, 13, 23, 23, 24, 17, 35, 19, 39, 31, 35, 23, 57, 25, 41, 36, 53, 29, 71, 31, 56, 47, 53, 47, 79, 37, 59, 55, 87, 41, 95, 43, 81, 74, 71, 47, 117, 49, 87, 71, 95, 53, 116, 71, 117, 79, 89, 59, 165, 61, 95, 100, 112, 83, 143, 67, 123, 95, 143, 71, 183, 73, 113, 118, 137, 95, 167, 79, 179, 108, 125, 83

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

nonn

Inverse Möbius transform of n+n^(1/2)*((-1)^tau(n)-1)/2. - Wesley Ivan Hurt, Jul 07 2025

			a(16) = 24; The divisors of 16 whose square does not divide 16 are 8 and 16. The sum of the divisors is then 8 + 16 = 24.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005 (tau), A000203 (sigma), A056595, A069290.

Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
sdnd[n_]:=Total[Select[Divisors[n],Mod[n,#^2]!=0&]]; Array[sdnd,100] (* Harvey P. Dale, Jul 07 2025 *)

a(n) = sumdiv(n, d, if (n % d^2, d)); \\ Michel Marcus, Jun 13 2021

from math import prod
from sympy import factorint
def A345320(n):
    f = factorint(n).items()
    return (prod(p**(q+1)-1 for p, q in f) - prod(p**(q//2+1)-1 for p, q in f))//prod(p-1 for p, q in f) # Chai Wah Wu, Jun 14 2021

a(n) = Sum_{k=1..n} k * (ceiling(n/k^2) - floor(n/k^2)) * (1 - ceiling(n/k) + floor(n/k)).
a(n) = A000203(n) - A069290(n). - Rémy Sigrist, Jun 14 2021
a(n) = Sum_{d|n} (d+d^(1/2)*((-1)^tau(d)-1)/2). - Wesley Ivan Hurt, Jul 07 2025

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

Original entry on oeis.org

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).

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

Original entry on oeis.org

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).

A365336 The sum of exponentially odd divisors of the square root of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 11, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A295295 at n = 64.
The sum of divisors of the square root of the largest square dividing n is A069290(n).
The number of these divisors is A365335(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000188, A005117, A033634, A069290, A295295, A365335.

f[p_, e_] := (p^(2*Floor[(e+2)/4] + 1) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(2*((f[i,2]+2)\4) + 1) - f[i,1])/(f[i,1]^2 - 1) + 1);}

a(n) = A033634(A000188(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(2*floor((e+2)/4) + 1) - p)/(p^2 - 1) + 1. [corrected by Georg Fischer, Oct 07 2023]
Dirichlet g.f.: zeta(s) * zeta(4*s-2) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(4*s-2)).
From Vaclav Kotesovec, Sep 02 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(4*s-2) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(5*s-2)).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(4*s-2) * Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)).
Let f(s) = Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)), then Sum_{k=1..n} a(k) ~ Pi^2/12 * n * (f(1) * (log(n) + 3*gamma - 1 + 24*zeta'(2)/Pi^2) + f'(1)), where f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444021876..., f'(1) = f(1) * Sum_{primes p} 2*(4*p-3)*log(p) /  (p^3 - 2*p + 1) = 1.617322217899181826790... and gamma is the Euler-Mascheroni constant A001620. (End)

A333694 Expansion of Sum_{k>=1} k * x^k / (1 - x^(k^2)).

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 8, 9, 10, 13, 12, 16, 14, 17, 16, 17, 18, 21, 20, 25, 25, 25, 24, 25, 26, 29, 28, 29, 30, 41, 32, 33, 34, 37, 36, 41, 38, 41, 43, 41, 42, 51, 44, 45, 46, 49, 48, 52, 50, 53, 52, 57, 54, 57, 61, 64, 61, 61, 60, 61, 62, 65, 64, 65, 66, 72, 68, 73, 70, 73

Offset: 1

Ilya Gutkovskiy, Apr 04 2020

nonn

Sum of divisors d of n such that n/d == 1 (mod d).

Cf. A000203, A069290, A163671.

a:= n-> add(`if`(irem(n/d-1, d)=0, d, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 04 2020

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, Mod[n/# - 1, #] == 0 &], {n, 1, 70}]

A333694(n) = sumdiv(n,d,d*(0==(((n/d)-1)%d))); \\ Antti Karttunen, Apr 04 2020, after the second Mathematica program.

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

A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 10, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, ... = n^(2/3)*A010057(n).

Winston de Greef, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=2,k=3,n).

Cf. A035316, A069290, A333843.

gsigma := proc(n,z,k)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n,d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n,2,3),n=1..80) ;

f[p_, e_] := (p^(2*(Floor[e/3] + 1)) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 24 2023 *)

a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ Michel Marcus, Mar 24 2023

for(n=1, 100, print1(direuler(p=2, n, (1/((1-X)*(1-p^2*X^3))))[n], ", ")) \\ Vaclav Kotesovec, Jun 26 2024

from math import prod
from sympy import factorint
def A361793(n): return prod((p**(e//3+1<<1)-1)//(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^3|n} d^2.
Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^2 * x^(k^3) / (1 - x^(k^3)). - Ilya Gutkovskiy, Jun 05 2024
From Vaclav Kotesovec, Jun 26 2024: (Start)
Dirichlet g.f.: zeta(s)*zeta(3*s-2).
Sum_{k=1..n} a(k) ~ n*(log(n) + 4*gamma - 1)/3, where gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

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

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A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

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A365332 The sum of divisors of the largest square dividing n.

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A380325 The sum of the square roots of the squares that divide the n-th exponentially odd number.

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A345320 Sum of the divisors of n whose square does not divide n.

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

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

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