A046951 -id:A046951 - cp's OEIS frontend

A252895 Numbers with an odd number of square divisors.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Walker Dewey Anderson, Mar 22 2015

nonn

Open lockers in the locker problem where the student numbers are the set of perfect squares.
The locker problem is a classic mathematical problem. Imagine a row containing an infinite number of lockers numbered from one to infinity. Also imagine an infinite number of students numbered from one to infinity. All of the lockers begin closed. The first student opens every locker that is a multiple of one, which is every locker. The second student closes every locker that is a multiple of two, so all of the even-numbered lockers are closed. The third student opens or closes every locker that is a multiple of three. This process continues for all of the students. [This is sometimes called the light switch problem - see A360845.]
A variant on the locker problem is when not all student numbers are considered; in the case of this sequence, only the square-numbered students open and close lockers. The sequence here is a list of the open lockers after all of the students have gone.
n is in the sequence if and only if it is the product of a squarefree number (A005117) and a fourth power (A000583). - Robert Israel, Apr 07 2015
Let D be the multiset containing d0(k), the divisor counting function, for each divisor k of n. n is in the sequence if and only if D admits a partition into two parts A and B such that the sum of the elements of A is exactly one more or less than the sum of the elements of B. For example, if n = 80, we have D = {1, 2, 2, 3, 4, 4, 5, 6, 8, 10}, and A = {1, 2, 3, 4, 4, 8} and B = {2, 5, 6, 10}. The sum of A is 22, and the sum of B is 23. - Griffin N. Macris, Oct 10 2016
From Amiram Eldar, Jul 07 2020: (Start)
Numbers k such that the largest square dividing k (A008833) is a fourth power.
The asymptotic density of this sequence is Pi^2/15 = A182448 = 0.657973... (Cesàro, 1885). (End)
Closed under the binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. - Peter Munn, Aug 01 2020

			The set of divisors of 6 is {1,2,3,6}, which contains only one perfect square: 1; therefore 6 is a term.
The set of divisors of 16 is {1,2,4,8,16}, which contains three perfect squares: 1, 4, and 16; therefore 16 is a term.
The set of divisors of 4 is {1,2,4}, which contains two perfect squares: 1 and 4; therefore 4 is not a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ernest Cesàro, Le plus grand diviseur carré, Annali di Matematica Pura ed Applicata, Vol. 13, No. 1 (1885), pp. 251-268, entire volume.
K. A. P. Dagal, Generalized Locker Problem, arXiv:1307.6455 [math.NT], 2013.
B. Torrence and S. Wagon, The Locker Problem, Crux Mathematicorum, 2007, 33(4), 232-236.

Cf. A000290, A000583, A005117, A008833, A046951, A182448, A252849, A360845.

Positions of ones in A335324.

a252895 n = a252895_list !! (n-1)
a252895_list = filter (odd . a046951) [1..]
-- Reinhard Zumkeller, Apr 06 2015

N:= 1000: # to get all terms <= N
S:= select(numtheory:-issqrfree, {$1..N}):
map(s -> seq(s*i^4, i = 1 .. floor((N/s)^(1/4))), S);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Apr 07 2015

Position[Length@ Select[Divisors@ #, IntegerQ@ Sqrt@ # &] & /@ Range@ 70, Integer?OddQ] // Flatten (* _Michael De Vlieger, Mar 23 2015 *)
a[n_] := DivisorSigma[0, Total[EulerPhi/@Select[Sqrt[Divisors[n]], IntegerQ]]]; Flatten[Position[a/@Range@100,?OddQ]] (* _Ivan N. Ianakiev, Apr 07 2015 *)
Select[Range@ 100, OddQ@ Length@ DeleteCases[Divisors@ #, k_ /; ! IntegerQ@ Sqrt@ k] &] (* Michael De Vlieger, Oct 10 2016 *)

isok(n) = sumdiv(n, d, issquare(d)) % 2; \\ Michel Marcus, Mar 22 2015

[n for n in [1..200] if len([x for x in divisors(n) if is_square(x)])%2==1] # Tom Edgar, Mar 22 2015

A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 28, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 56, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 84, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 120, 65, 66, 67

Offset: 1

Werner Schulte, Jan 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A069290, A055615, A037213.
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =
0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 20 2021

f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2021 *)

A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ Antti Karttunen, Aug 19 2021

Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).
Dirichlet convolution of A000010 and A069290.
Dirichlet convolution with A055615 equals A037213.
G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021
Sum_{k=1..n} a(k) ~ zeta(3)*n^2/2. - Vaclav Kotesovec, Aug 19 2021
a(n) = n * Sum_{d^2|n} 1/d. - Wesley Ivan Hurt, Feb 14 2022

A351309 Sum of the 4th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 257, 1, 1, 1, 65793, 1, 6562, 1, 257, 1, 1, 1, 257, 390626, 1, 6562, 257, 1, 1, 1, 65793, 1, 1, 1, 1686434, 1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 65793, 5764802, 390626, 1, 257, 1, 6562, 1, 257, 1, 1, 1, 257, 1, 1, 6562, 16843009, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^4 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 29 2024

			a(16) = 65793; a(16) = Sum_{d^2|16} (d^2)^4 = (1^2)^4 + (2^2)^4 + (4^2)^4 = 65793.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), this sequence (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A001159.

f[p_, e_] := (p^(8*(1 + Floor[e/2])) - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^4],{n,70}] (* Harvey P. Dale, Feb 11 2023 *)

a(n) = sumdiv(n, d, if (issquare(d), d^4)); \\ Michel Marcus, Jun 05 2024

a(n) = Sum_{d^2|n} (d^2)^4.
Multiplicative with a(p) = (p^(8*(1+floor(e/2))) - 1)/(p^8 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-8).
Sum_{k=1..n} a(k) ~ (zeta(9/2)/9) * n^(9/2). (End)
G.f.: Sum_{k>=1} k^8 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024
a(n) = Sum_{d|n} d^4 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 29 2024
a(n) = Sum_{d|n} lambda(d)*d^4*sigma_4(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351310 Sum of the 5th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1025, 1, 1, 1, 1049601, 1, 59050, 1, 1025, 1, 1, 1, 1025, 9765626, 1, 59050, 1025, 1, 1, 1, 1049601, 1, 1, 1, 60526250, 1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1049601, 282475250, 9765626, 1, 1025, 1, 59050, 1, 1025, 1, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^5 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 1049601; a(16) = Sum_{d^2|16} (d^2)^5 = (1^2)^5 + (2^2)^5 + (4^2)^5 = 1049601.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), this sequence (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A001160.

f[p_, e_] := (p^(10*(1 + Floor[e/2])) - 1)/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

a(n) = Sum_{d^2|n} (d^2)^5.
Multiplicative with a(p) = (p^(10*(1+floor(e/2))) - 1)/(p^10 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-10).
Sum_{k=1..n} a(k) ~ (zeta(11/2)/11) * n^(11/2). (End)
a(n) = Sum_{d|n} d^5 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^5*sigma_5(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351311 Sum of the 6th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 4097, 1, 1, 1, 16781313, 1, 531442, 1, 4097, 1, 1, 1, 4097, 244140626, 1, 531442, 4097, 1, 1, 1, 16781313, 1, 1, 1, 2177317874, 1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 16781313, 13841287202, 244140626, 1, 4097, 1, 531442, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^6 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 16781313; a(16) = Sum_{d^2|16} (d^2)^6 = (1^2)^6 + (2^2)^6 + (4^2)^6 = 16781313.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), this sequence (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A013954.

f[p_, e_] := (p^(12*(1 + Floor[e/2])) - 1)/(p^12 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

a(n) = Sum_{d^2|n} (d^2)^6.
Multiplicative with a(p) = (p^(12*(1+floor(e/2))) - 1)/(p^12 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-12).
Sum_{k=1..n} a(k) ~ (zeta(13/2)/13) * n^(13/2). (End)
a(n) = Sum_{d|n} d^6 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^6*sigma_6(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351313 Sum of the 7th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 16385, 1, 1, 1, 268451841, 1, 4782970, 1, 16385, 1, 1, 1, 16385, 6103515626, 1, 4782970, 16385, 1, 1, 1, 268451841, 1, 1, 1, 78368963450, 1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 268451841, 678223072850, 6103515626, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^7 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 268451841; a(16) = Sum_{d^2|16} (d^2)^7 = (1^2)^7 + (2^2)^7 + (4^2)^7 = 268451841.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), this sequence (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A013955.

f[p_, e_] := (p^(14*(1 + Floor[e/2])) - 1)/(p^14 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

a(n) = Sum_{d^2|n} (d^2)^7.
Multiplicative with a(p) = (p^(14*(1+floor(e/2))) - 1)/(p^14 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-14).
Sum_{k=1..n} a(k) ~ (zeta(15/2)/15) * n^(15/2). (End)
a(n) = Sum_{d|n} d^7 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^7*sigma_7(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351314 Sum of the 8th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 65537, 1, 1, 1, 4295032833, 1, 43046722, 1, 65537, 1, 1, 1, 65537, 152587890626, 1, 43046722, 65537, 1, 1, 1, 4295032833, 1, 1, 1, 2821153019714, 1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 4295032833, 33232930569602, 152587890626

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^8 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), this sequence (k=8), A351315 (k=9), A351316 (k=10).

Cf. A010052, A008836, A013956.

f[p_, e_] := (p^(16*(1 + Floor[e/2])) - 1)/(p^16 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^8],{n,80}] (* Harvey P. Dale, Feb 13 2022 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^16*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

a(n) = Sum_{d^2|n} (d^2)^8.
Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^16*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-16).
Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)
a(n) = Sum_{d|n} d^8 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^8*sigma_8(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351315 Sum of the 9th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 262145, 1, 1, 1, 68719738881, 1, 387420490, 1, 262145, 1, 1, 1, 262145, 3814697265626, 1, 387420490, 262145, 1, 1, 1, 68719738881, 1, 1, 1, 101560344351050, 1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 68719738881

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^9 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 68719738881; a(16) = Sum_{d^2|16} (d^2)^9 = (1^2)^9 + (2^2)^9 + (4^2)^9 = 68719738881.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), this sequence (k=9), A351316 (k=10).

Cf. A010052, A008836, A013957.

f[p_, e_] := (p^(18*(1 + Floor[e/2])) - 1)/(p^18 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
snp[n_]:=Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^9]; Array[snp,50] (* Harvey P. Dale, May 25 2025 *)

a(n) = Sum_{d^2|n} (d^2)^9.
Multiplicative with a(p) = (p^(18*(1+floor(e/2))) - 1)/(p^18 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-18).
Sum_{k=1..n} a(k) ~ (zeta(19/2)/19) * n^(19/2). (End)
a(n) = Sum_{d|n} d^9 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^9*sigma_9(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A351600 a(n) = n^2 * Sum_{d^2|n} 1 / d^2.

Original entry on oeis.org

1, 4, 9, 20, 25, 36, 49, 80, 90, 100, 121, 180, 169, 196, 225, 336, 289, 360, 361, 500, 441, 484, 529, 720, 650, 676, 810, 980, 841, 900, 961, 1344, 1089, 1156, 1225, 1800, 1369, 1444, 1521, 2000, 1681, 1764, 1849, 2420, 2250, 2116, 2209, 3024, 2450, 2600, 2601, 3380, 2809

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), this sequence (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013662, A076752.

f[p_, e_] := p^2*(p^(2*e) - p^(2*Floor[(e - 1)/2]))/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^2*sumdiv(n, d, if (issquare(d), 1/d)); \\ Michel Marcus, Feb 15 2022

G.f.: Sum_{k>=1} k^2 * x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3. - Ilya Gutkovskiy, Feb 21 2022
Multiplicative with a(p^e) = p^2*(p^(2*e) - p^(2*floor((e-1)/2)))/(p^2 - 1). - Sebastian Karlsson, Feb 25 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(4)/3 = Pi^4/270 = 0.360774... . - Amiram Eldar, Nov 13 2022

A351601 a(n) = n^3 * Sum_{d^2|n} 1 / d^3.

Original entry on oeis.org

1, 8, 27, 72, 125, 216, 343, 576, 756, 1000, 1331, 1944, 2197, 2744, 3375, 4672, 4913, 6048, 6859, 9000, 9261, 10648, 12167, 15552, 15750, 17576, 20412, 24696, 24389, 27000, 29791, 37376, 35937, 39304, 42875, 54432, 50653, 54872, 59319, 72000, 68921, 74088, 79507, 95832

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), this sequence (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013663.

f[p_, e_] := p^3*(p^(3*e) - p^(3*Floor[(e - 1)/2]))/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^3*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^3))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^3*(p^(3*e) - p^(3*floor((e-1)/2)))/(p^3 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(5)/4 = 0.259231... . - Amiram Eldar, Nov 13 2022

cp's OEIS Frontend

A252895 Numbers with an odd number of square divisors.

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A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

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A351309 Sum of the 4th powers of the square divisors of n.

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A351310 Sum of the 5th powers of the square divisors of n.

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A351311 Sum of the 6th powers of the square divisors of n.

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A351313 Sum of the 7th powers of the square divisors of n.

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A351314 Sum of the 8th powers of the square divisors of n.

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