A351270 -id:A351270 - cp's OEIS frontend

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.
Also the number of squarefree divisors of n. - Labos Elemer
Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
Also shadow transform of pronic numbers A002378.
For n >= 1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n. a(n) is the rank of A. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003
a(n) is also the number of solutions to x^2 - x == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
a(n) is the number of squarefree divisors of n, but in general the set of unitary divisors of n is not the set of squarefree divisors (compare the rows of A077610 and A206778). - Jaroslav Krizek, May 04 2009
Row lengths of the triangles in A077610 and in A206778. - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. - Michel Lagneau, Nov 15 2012
a(n) is the number of irreducible fractions y/x that satisfy x*y=n (and gcd(x,y)=1), x and y positive integers. - Luc Rousseau, Jul 09 2017
a(n) is the number of (x,y) lattice points satisfying both x*y=n and (x,y) is visible from (0,0), x and y positive integers. - Luc Rousseau, Jul 10 2017
Conjecture: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the unitary divisors of n is divisible by the sum of the k-th powers of the odd unitary divisors of n (note that this sequence lists the sum of the 0th powers of the unitary divisors of n). - Ivan N. Ianakiev, Feb 18 2018
a(n) is the number of one-digit numbers, k, when written in base n such that k and k^2 end in the same digit. - Matthew Scroggs, Jun 01 2018
Dirichlet convolution of A271102 and A000005. - Vaclav Kotesovec, Apr 08 2019
Conjecture: Let b(i; n), n > 0, be multiplicative sequences for some fixed integer i >= 0 with b(i; p^e) = (Sum_{k=1..i+1} A164652(i, k) * e^(k-1)) * (i+2) / (i!) for prime p and e > 0. Then we have Dirichlet generating functions: Sum_{n > 0} b(i; n) / n^s = (zeta(s))^(i+2) / zeta((i+2) * s). Examples for i=0 this sequence, for i=1 A226602, and for i=2 A286779. - Werner Schulte, Feb 17 2022
The smallest integer with 2^m unitary divisors, or equivalently, the smallest integer with 2^m squarefree divisors, is A002110(m). - Bernard Schott, Oct 04 2022

			a(12) = 4 because the four unitary divisors of 12 are 1, 3, 4, 12, and also because the four squarefree divisors of 12 are 1, 2, 3, 6.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
Jon Maiga, Upper bound of Fibonacci entry points, 2019.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.
Index entries for sequences computed from exponents in factorization of n

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A002110, A034448, A047994, A061142, A277561.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: this sequence (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: this sequence (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).
Cf. A020821 (Dgf at s=2), A177057 (Dgf at s=4).

a034444 = length . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

[#[d:d in Divisors(n)|Gcd(d,n div d) eq 1]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

[&+[Abs(MoebiusMu(d)):d in Divisors(n)]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od:
with(numtheory);
# returns the number of unitary divisors of n and a list of them
f:=proc(n)
local ct,i,t1,ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;
od:
RETURN([ct,ans]);
end;
# N. J. A. Sloane, May 01 2013
# alternative Maple program:
a:= n-> 2^nops(ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Jan 23 2024
a := n -> 2^NumberTheory:-NumberOfPrimeFactors(n, distinct):  # Peter Luschny, May 13 2025

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* Jean-François Alcover, Apr 05 2011 *)
Table[2^PrimeNu[n],{n,110}] (* Harvey P. Dale, Jul 14 2011 *)

a(n)=1<Charles R Greathouse IV, Feb 11 2011

for(n=1, 100, print1(direuler(p=2, n, (1+X)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

from sympy import divisors, gcd
def a(n):
    return sum(1 for d in divisors(n) if gcd(d, n//d)==1)
# Indranil Ghosh, Apr 16 2017

from sympy import primefactors
def a(n): return 2**len(primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

(define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; Antti Karttunen, May 29 2017

a(n) = Sum_{d|n} abs(mu(n)) = 2^(number of different primes dividing n) = 2^A001221(n), with mu(n) = A008683(n). [Added Möbius formula. - Wolfdieter Lang, Jan 11 2020]
a(n) = Product_{ primes p|n } (1 + Legendre(1, p)).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley, Oct 25 2001
a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - Benoit Cloitre, Oct 03 2002
Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005
Asymptotically [Finch] the cumulative sum of a(n) = Sum_{n=1..N} a(n) ~ (6/(Pi^2))*N*log(N) + (6/(Pi^2))*(2*gamma - 1 - (12/(Pi^2))*zeta'(2))*N + O(sqrt(N)). - Jonathan Vos Post, May 08 2005 [typo corrected by Vaclav Kotesovec, Sep 13 2018]
a(n) = Sum_{d|n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). - Enrique Pérez Herrero, Nov 13 2009
a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013
G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014
a(n) = Sum_{d|n} lambda(d)*mu(d), where lambda is A008836. - Enrique Pérez Herrero, Apr 27 2014
a(n) = A277561(A156552(n)). - Antti Karttunen, May 29 2017
a(n) = A005361(n^2)/A005361(n). - Velin Yanev, Jul 26 2017
L.g.f.: -log(Product_{k>=1} (1 - mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - Torlach Rush, Jan 20 2020
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - Amiram Eldar, May 29 2020
a(n) = lim_{k->oo} A000005(n^(2*k))/A000005(n^k). - Velin Yanev and Amiram Eldar, Jan 10 2025

More terms from James Sellers, Jun 20 2000

A048250 Sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 3, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 3, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 12, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 3, 84, 144, 68, 54, 96, 144, 72, 12, 74

Offset: 1

Labos Elemer

Also sum of divisors of the squarefree kernel of n: a(n) = A000203(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
The absolute values of the Dirichlet inverse of A001615. - R. J. Mathar, Dec 22 2010
Row sums of the triangle in A206778. - Reinhard Zumkeller, Feb 12 2012
Inverse Möbius transform of n * mu(n)^2 = |A055615(n)|. - Wesley Ivan Hurt, Jun 08 2023

			For n=1000, out of the 16 divisors, four are squarefree: {1,2,5,10}. Their sum is 18. Or, 1000 = 2^3*5^3 hence a(1000) = (2+1)*(5+1) = 18.

D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972) 65-74.

T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Index entries for sequences related to sums of squares

Cf. A003557, A007913, A007947, A023900, A034448, A206787, A309192.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), this sequence (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Cf. A240976 (tenth of Dgf at s=3).

a034448 = sum . a206778_row  -- Reinhard Zumkeller, Feb 12 2012

A048250 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]): od: RETURN(ans) end:
# alternative:
seq(mul(1+p, p = numtheory:-factorset(n)), n=1..1000); # Robert Israel, Mar 18 2015

sumOfSquareFreeDivisors[ n_ ] := Plus @@ Select[ Divisors[ n ], MoebiusMu[ # ] != 0 & ]; Table[ sumOfSquareFreeDivisors[ i ], {i, 85} ]
Table[Total[Select[Divisors[n],SquareFreeQ]],{n,80}] (* Harvey P. Dale, Jan 25 2013 *)
a[1] = 1; a[n_] := Times@@(1 + FactorInteger[n][[;;,1]]); Array[a, 100] (* Amiram Eldar, Dec 19 2018 *)

a(n)=if(n<1,0,sumdiv(n,d,if(core(d)==d,d)))

a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-X))[n])

a(n)=sumdiv(n,d,moebius(d)^2*d); \\ Joerg Arndt, Jul 06 2011

a(n)=my(f=factor(n)); for(i=1,#f~,f[i,2]=1); sigma(f) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A048250(n): return prod(p+1 for p in primefactors(n)) # Chai Wah Wu, Apr 20 2023

def A048250(n): return mul(map(lambda p: p+1, prime_divisors(n)))
[A048250(n) for n in (1..73)]  # Peter Luschny, May 23 2013

If n = Product p_i^e_i, a(n) = Product (p_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s-2). - Michael Somos, Sep 08 2002
a(n) = Sum_{d|n} mu(d)^2*d = Sum_{d|n} |A055615(d)|. - Benoit Cloitre, Dec 09 2002
Pieter Moree (moree(AT)mpim-bonn.mpg.de), Feb 20 2004 can show that Sum_{n <= x} a(n) = x^2/2 + O(x*sqrt{x}) and adds: "As S. R. Finch pointed out to me, in Suryanarayana's paper this is proved under the Riemann hypothesis with error term O(x^{7/5+epsilon})".
a(n) = psi(rad(n)) = A001615(A007947(n)). - Enrique Pérez Herrero, Aug 24 2010
a(n) = rad(n)*psi(n)/n = A001615(n)*A007947(n)/n. - Enrique Pérez Herrero, Aug 31 2010
G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = 1. - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d divides n} mu(d)^2*core(d), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

A351265 Sum of the squares of the squarefree divisors of n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 5, 10, 130, 122, 50, 170, 250, 260, 5, 290, 50, 362, 130, 500, 610, 530, 50, 26, 850, 10, 250, 842, 1300, 962, 5, 1220, 1450, 1300, 50, 1370, 1810, 1700, 130, 1682, 2500, 1850, 610, 260, 2650, 2210, 50, 50, 130, 2900, 850, 2810, 50, 3172, 250, 3620

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^2 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(6) = 50; a(6) = Sum_{d|6} d^2 * mu(d)^2 = 1^2*1 + 2^2*1 + 3^2*1 + 6^2*1 = 50.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A253905, A328639, A322360.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), this sequence (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^2); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^2],{n,80}] (* Harvey P. Dale, Dec 26 2024 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^2)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^2 * mu(d)^2.
a(n) = abs(A328639(n)).
G.f.: Sum_{k>=1} mu(k)^2 * k^2 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^2. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(3*zeta(2)) = A253905 / 3 = 0.243587... . - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s-4). - Michael Shamos, Aug 05 2023

A351266 Sum of the cubes of the squarefree divisors of n.

Original entry on oeis.org

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 9, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 9, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512, 103824

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^3 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 9; a(4) = Sum_{d|4} d^3 * mu(d)^2 = 1^3*1 + 2^3*1 + 4^3*0 = 9.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013662, A328640.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), this sequence (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^3); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^3)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^3 * mu(d)^2.
a(n) = abs(A328640(n)).
G.f.: Sum_{k>=1} mu(k)^2 * k^3 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^3. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(4)/(4*zeta(2)) = Pi^2/60 = 0.164493... . - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-6). - Michael Shamos, Feb 09 2025

A351267 Sum of the 4th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 17, 82, 17, 626, 1394, 2402, 17, 82, 10642, 14642, 1394, 28562, 40834, 51332, 17, 83522, 1394, 130322, 10642, 196964, 248914, 279842, 1394, 626, 485554, 82, 40834, 707282, 872644, 923522, 17, 1200644, 1419874, 1503652, 1394, 1874162, 2215474, 2342084, 10642, 2825762, 3348388

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^4 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 17; a(4) = Sum_{d|4} d^4 * mu(d)^2 = 1^4*1 + 2^4*1 + 4^4*0 = 17.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013663.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), this sequence (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^4); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^4)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^4 * mu(d)^2.
G.f.: Sum_{k>=1} mu(k)^2 * k^4 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^4. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Nov 10 2022

A351268 Sum of the 5th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 33, 244, 33, 3126, 8052, 16808, 33, 244, 103158, 161052, 8052, 371294, 554664, 762744, 33, 1419858, 8052, 2476100, 103158, 4101152, 5314716, 6436344, 8052, 3126, 12252702, 244, 554664, 20511150, 25170552, 28629152, 33, 39296688, 46855314, 52541808, 8052, 69343958

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^5 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 33; a(4) = Sum_{d|4} d^5 * mu(d)^2 = 1^5*1 + 2^5*1 + 4^4*0 = 33.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013664.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), this sequence (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^5); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^5 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^5. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^5 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(6)/(6*zeta(2)) = Pi^4/945 = 0.103078... . - Amiram Eldar, Nov 10 2022

A351269 Sum of the 6th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 65, 730, 65, 15626, 47450, 117650, 65, 730, 1015690, 1771562, 47450, 4826810, 7647250, 11406980, 65, 24137570, 47450, 47045882, 1015690, 85884500, 115151530, 148035890, 47450, 15626, 313742650, 730, 7647250, 594823322, 741453700, 887503682, 65, 1293240260

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^6 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 65; a(4) = Sum_{d|4} d^6 * mu(d)^2 = 1^6*1 + 2^6*1 + 4^6*0 = 65.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013665.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), this sequence (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^6); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^6 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^6. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^6 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/(7*zeta(2)) = 0.0875718... . - Amiram Eldar, Nov 10 2022

A351271 Sum of the 8th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 257, 6562, 257, 390626, 1686434, 5764802, 257, 6562, 100390882, 214358882, 1686434, 815730722, 1481554114, 2563287812, 257, 6975757442, 1686434, 16983563042, 100390882, 37828630724, 55090232674, 78310985282, 1686434, 390626, 209642795554, 6562, 1481554114, 500246412962

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^8 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 257; a(4) = Sum_{d|4} d^8 * mu(d)^2 = 1^8*1 + 2^8*1 + 4^8*0 = 257.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013667.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), this sequence (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^8); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

a(n) = Sum_{d|n} d^8 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^8. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^8 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(9*zeta(2)) = 0.0676831... . - Amiram Eldar, Nov 10 2022

A351272 Sum of the 9th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 513, 19684, 513, 1953126, 10097892, 40353608, 513, 19684, 1001953638, 2357947692, 10097892, 10604499374, 20701400904, 38445332184, 513, 118587876498, 10097892, 322687697780, 1001953638, 794320419872, 1209627165996, 1801152661464, 10097892, 1953126, 5440108178862

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^9 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 513; a(4) = Sum_{d|4} d^9 * mu(d)^2 = 1^9*1 + 2^9*1 + 4^9*0 = 513.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013668.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), this sequence (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^9); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^9],{n,30}] (* Harvey P. Dale, Feb 21 2023 *)

a(n) = Sum_{d|n} d^9 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^9. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^9 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/(10*zeta(2)) = Pi^8/155925 = 0.0608531... . - Amiram Eldar, Nov 10 2022

A351273 Sum of the 10th powers of the squarefree divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1025, 9765626, 60526250, 282475250, 1025, 59050, 10009766650, 25937424602, 60526250, 137858491850, 289537131250, 576660215300, 1025, 2015993900450, 60526250, 6131066257802, 10009766650, 16680163512500, 26585860217050, 41426511213650, 60526250

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^10 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(4) = 1025; a(4) = Sum_{d|4} d^10 * mu(d)^2 = 1^10*1 + 2^10*1 + 4^10*0 = 1025.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A013661, A013669.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), this sequence (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^10); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^10],{n,25}] (* Harvey P. Dale, Nov 20 2022 *)

a(n) = Sum_{d|n} d^10 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^10. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^10 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/(11*zeta(2)) = 0.0552934... . - Amiram Eldar, Nov 10 2022

cp's OEIS Frontend

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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A048250 Sum of the squarefree divisors of n.

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A351265 Sum of the squares of the squarefree divisors of n.

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A351266 Sum of the cubes of the squarefree divisors of n.

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A351267 Sum of the 4th powers of the squarefree divisors of n.

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A351268 Sum of the 5th powers of the squarefree divisors of n.