A046951 -id:A046951 - cp's OEIS frontend

A130279 Smallest number having exactly n square divisors.

1, 4, 16, 36, 256, 144, 4096, 576, 1296, 2304, 1048576, 3600, 16777216, 36864, 20736, 14400, 4294967296, 32400, 68719476736, 57600, 331776, 9437184, 17592186044416, 129600, 1679616, 150994944, 810000, 921600, 72057594037927936

Offset: 1

Reinhard Zumkeller, May 20 2007

nonn

A046951(a(n)) = n and A046951(m) <> n for m < a(n);

all terms are smooth squares: if prime(k) is a factor of a(n) then also prime(i) are factors, i

a(p) = 2^(2*(p-1)) for primes p;

if prime(j) is the greatest prime factor of a(n) then a(2*n) = a(n)*prime(j+1)^2;

A001221(a(n)) = A122375(n); A001222(a(n)) = 2*A122376(n).

a(n+1) is the smallest nonsquarefree number m such that Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m has exactly n solutions, for n >= 0 (A353282); example: a(4) = 36 and 36 is the smallest number m such that equation S(x,y) = m has exactly 3 solutions: (9,1), (8,2), (5,5). - Bernard Schott, Apr 13 2022

a(n) is the square of the smallest integer having exactly n divisors (see formula with proof). - Bernard Schott, Oct 01 2022

Amiram Eldar, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A001221, A005179, A046951, A046952, A122375, A122376, A353282.

Cf. A357450 (similar, but with odd squares divisors).

a(n) = my(k=1); while(sumdiv(k, d, issquare(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2019

From Bernard Schott, Oct 01 2022: (Start)
a(n) = A005179(n)^2.
Proof: Suppose a(n) = Product p_i^(2*e_i), where the p_i are primes. Then the n square divisors are all of the form d = Product p_i^(2*k_i) with 0 <= k_i <= e_i. As a(n) = Product (p_i^e_i)^2 = (Product (p_i^e_i))^2, we get that sqrt(a(n)) = Product (p_i^e_i). This is the prime decomposition of sqrt(a(n)). As there is a bijection between prime factors p_i^(2*k_i) and (p_i^k_i), there is also bijection between square divisors of a(n) and divisors of sqrt(a(n)). We conclude that sqrt(a(n)) is the smallest integer that has exactly n divisors. (End)

A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 4, 8

Offset: 1

Labos Elemer, Jun 13 2000

Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
a(n) = 1 iff n is a square. - Bernard Schott, Oct 22 2019
a(n) is the number of the unitary divisors (cf. A077610) of n that are exponentially odd (A268335). - Amiram Eldar, Nov 11 2022
The number of infinitary divisors of n that are squarefree (A005117). - Amiram Eldar, Jan 09 2024

			n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A005117, A007913, A034444, A077610, A162642, A268335, A295316, A350389.

with(numtheory):
a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 21 2015

a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
Array[a, 105] (* Jean-François Alcover, Mar 28 2017 *)
f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)

A055076(n) = if(1==n,n,my(es=factor(n)[,2]~); prod(i=1,#es,2^(es[i]%2))); \\ Antti Karttunen, Apr 05 2021

;; With memoization-macro definec.
(definec (A055076 n) (if (= 1 n) n (* (+ 1 (A000035 (A067029 n))) (A055076 (A028234 n))))) ;; Antti Karttunen, Dec 02 2017

Multiplicative with a(p^e) = 2^(e mod 2). - Vladeta Jovovic, Dec 13 2002
a(n) = 2^A162642(n). - Antti Karttunen, Dec 02 2017
a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - Antti Karttunen, Apr 05 2021
From Amiram Eldar, Sep 09 2023: (Start)
a(n) = A034444(A350389(n)).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
and gamma is the Euler-Mascheroni constant A001620. (End)

A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

Original entry on oeis.org

16, 32, 36, 48, 64, 72, 80, 81, 96, 100, 108, 112, 128, 144, 160, 162, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 243, 252, 256, 272, 288, 300, 304, 320, 324, 336, 352, 360, 368, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 464, 468, 480, 484

Offset: 1

Labos Elemer

nonn

Numbers n that are expressible as a product of 2 "nonsquarefree" numbers (i.e., there are 2 integers x,y in A001694 such that n = xy). - Benoit Cloitre, Jan 01 2003
Also numbers having more than one square divisor > 1: A046951(a(n)) > 2. - Reinhard Zumkeller, Apr 08 2003
The asymptotic density of this sequence is 1 - (6/Pi^2)*(1 + Sum_{n>=1} 1/prime(n)^2) = 1 - A059956 * (1 + A085548) = 0.1171394347594477824... . - Amiram Eldar, Sep 25 2022

			36 is in the sequence since the number of its unitary divisors, {1, 4, 9, 36} is 4 which is smaller than 5, the number of its non-unitary divisors, {2, 3, 6, 12, 18}.

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

Cf. A000005, A001221, A034444, A048105, A048109, A082293, A013929, A082294, A082295.

Cf. A085548, A059956.

Select[Range[484], DivisorSigma[0, #] > 2^(PrimeNu[#]+1) &] (* Amiram Eldar, Jun 11 2019 *)

is(n)=my(f=factor(n)[,2],t); for(i=1,#f,if(f[i]>1, if(t||f[i]>3, return(1), t=1))); 0 \\ Charles R Greathouse IV, Sep 17 2015

A000005(a(n)) > 2^(1 + A001221(a(n))).

A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 07 2019

First differs from A050361 at n = 64.
From Amiram Eldar, Sep 08 2023: (Start)
The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024

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

Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.

Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.

fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); Array[a, 100]

a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023

Multiplicative with a(p^e) = floor((e+1)/2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022
a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023
From Amiram Eldar, Sep 08 2023: (Start)
a(n) = A046951(A003557(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)

Name corrected by Amiram Eldar, Sep 08 2023

A351307 Sum of the squares of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 17, 1, 1, 1, 273, 1, 82, 1, 17, 1, 1, 1, 17, 626, 1, 82, 17, 1, 1, 1, 273, 1, 1, 1, 1394, 1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 273, 2402, 626, 1, 17, 1, 82, 1, 17, 1, 1, 1, 17, 1, 1, 82, 4369, 1, 1, 1, 17, 1, 1, 1, 1394, 1, 1, 626, 17, 1, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

			a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.

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), this sequence (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A247041 (zeta(5/2)), A008836, A001157.

f[p_, e_] := (p^(4*(1 + Floor[e/2])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

from math import prod
from sympy import factorint
def A351307(n): return prod((p**(4+((e&-2)<<1))-1)//(p**4-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 11 2024

a(n) = Sum_{d^2|n} (d^2)^2.
Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^4*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 19 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-4).
Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)
a(n) = Sum_{d|n} d^2 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d^2*sigma_2(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A351308 Sum of the cubes of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 65, 1, 1, 1, 4161, 1, 730, 1, 65, 1, 1, 1, 65, 15626, 1, 730, 65, 1, 1, 1, 4161, 1, 1, 1, 47450, 1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 4161, 117650, 15626, 1, 65, 1, 730, 1, 65, 1, 1, 1, 65, 1, 1, 730, 266305, 1, 1, 1, 65, 1, 1, 1, 47450, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^3 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

			a(16) = 4161; a(16) = Sum_{d^2|16} (d^2)^3 = (1^2)^3 + (2^2)^3 + (4^2)^3 = 4161.

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), this sequence (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A261804 (zeta(7/2)), A008836, A001158.

f[p_, e_] := (p^(6*(1 + Floor[e/2])) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

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

a(n) = Sum_{d^2|n} (d^2)^3.
Multiplicative with a(p) = (p^(6*(1+floor(e/2))) - 1)/(p^6 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 19 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-6).
Sum_{k=1..n} a(k) ~ (zeta(7/2)/7) * n^(7/2). (End)
G.f.: Sum_{k>=1} k^6 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024
a(n) = Sum_{d|n} d^3 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d^3*sigma_3(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A046952 Sets record for f(n) = |{(a,b):a*b=n and a|b}|. Also squares of highly composite numbers A002182.

Original entry on oeis.org

1, 4, 16, 36, 144, 576, 1296, 2304, 3600, 14400, 32400, 57600, 129600, 518400, 705600, 1587600, 2822400, 6350400, 25401600, 57153600, 101606400, 228614400, 406425600, 635040000, 768398400, 2057529600, 2540160000, 3073593600

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR automatic theory formation program.
Also, integers whose number of square divisors sets a new record. - Bernard Schott, Jan 14 2022
As a(n) is the square of n-th highly composite number (A002182), the record number of square divisors of a(n) is A046951(a(n)) = tau(A002182(n)) = A002183(n) where tau is the divisor counting function (A000005). - Bernard Schott, Jan 15 2022
Integers m for which number of solutions (A353282) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m sets a new record; these records are respectively 0, 1, 2, 3, 5, 7, ... Example: the 5 solutions for S(x,y) = 144 are (36,1), (32,2), (27,3), (20,5), (11,11). - Bernard Schott, Apr 19 2022

			f(1)=1, (first with 1), f(4)=2 (first with 2), f(16)=3 (first with 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A000005, A002182, A002183, A046951.

Cf. A350756 (similar, with triangular divisors).

a(n) = A002182(n)^2. - Bernard Schott, Jan 14 2022

A063775 Number of 4th powers dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Aug 16 2001

			a(79) = 1 since 79 is divisible by 1 = 1^4.
a(80) = 2 since 80 is divisible by 1 and 16 = 2^4.
a(81) = 2 since 81 is divisible by 1 and 81 = 3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Cf. A046951 (number of squares), A061704 (number of cubes).

Cf. A000005, A053164, A046951, A000188.

seq(coeff(series(add(x^(k^4)/(1-x^(k^4)),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Dec 29 2018

nn = 100;f[list_, i_] := list[[i]];
Table[DirichletConvolve[f[Boole[Map[IntegerQ[#] &, Map[#^(1/4) &,Range[nn]]]], n],f[Table[1, {nn}], n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 07 2015 *)
f[p_, e_] := 1 + Floor[e/4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

{ for (n=1, 2000, k=2; a=1; while ((p=k^4) <= n, if (n%p == 0, a++); k++); write("b063775.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 30 2009

a(n) = A000005(A053164(n)) = A046951(A000188(n)).
Multiplicative with a(p^e) = 1 + floor(e/4).
Dirichlet g.f.: zeta^2(4s)*Product_{primes p} (1 + p^(-s) + p^(-2s) + p^(-3s)). - R. J. Mathar, Jan 11 2012
G.f.: Sum_{k>=1} x^(k^4)/(1 - x^(k^4)). - Ilya Gutkovskiy, Mar 21 2017
Dirichlet g.f.: zeta(s) * zeta(4s). - Álvar Ibeas, Dec 29 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n / 90 + Zeta(1/4) * n^(1/4). - Vaclav Kotesovec, Feb 03 2019

A082293 Numbers having exactly one square divisor > 1.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 184, 188, 189, 198, 204, 207, 212

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by Peter Munn, Nov 17 2020]

The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - Amiram Eldar, Jul 07 2020

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

Cf. A005117, A046951, A222056.
Complement of A048111 within A013929.
Subsequence of A252849.
Disjoint union of A048109 and A060687.
A285508 is a subsequence.

Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* Amiram Eldar, Jul 07 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, primerange
def A082293(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x-sum(g(x//p**2) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A046951(a(n)) = 2.

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cp's OEIS Frontend

A013936 a(n) = Sum_{k=1..n} floor(n/k^2).

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A130279 Smallest number having exactly n square divisors.

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A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

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A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

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A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

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A351307 Sum of the squares of the square divisors of n.

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A351308 Sum of the cubes of the square divisors of n.

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