A351302 -id:A351302 - 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

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96

Offset: 1

N. J. A. Sloane

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L =  consists of all vectors of the form mV + nW, (m,n integers). A sublattice S =  has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>,  and  (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice  of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, Representing and counting the subgroups of the group Z_m X Z_n, arXiv:1211.1797 [math.GR], 2012.
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications 14:2 (2015), 73-88.
F. A. Lewis et al., Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825 [math.NT], 2010-2011.
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (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), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018

A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012

Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)

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

{a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013

a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021

def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

More terms from Olivier Gérard, Aug 15 1997

A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).

Original entry on oeis.org

1, 5, 10, 20, 26, 50, 50, 80, 90, 130, 122, 200, 170, 250, 260, 320, 290, 450, 362, 520, 500, 610, 530, 800, 650, 850, 810, 1000, 842, 1300, 962, 1280, 1220, 1450, 1300, 1800, 1370, 1810, 1700, 2080, 1682, 2500, 1850, 2440, 2340, 2650, 2210

Offset: 1

N. J. A. Sloane, Dec 08 2001

The sequence may be considered as psi_2, a generalization of Dedekind psi function, where psi_1 is A001615. - Enrique Pérez Herrero, Jul 06 2011

József Sándor, Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth 2002, pp. 193.

E. Pérez Herrero, Table of n, a(n) for n = 1..10000
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 Chapter 3.14.2

Cf. A000010, A007434, A156733, A301978, A301980, A321973.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), this sequence (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

A065958 := proc(n) local i,j,k,t1,t2,t3; t1 := ifactors(n)[2]; t2 := n^2*mul((1+1/(t1[i][1])^2),i=1..nops(t1)); end;

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A065958[n_]:=JordanTotient[n,4]/JordanTotient[n,2]; (* Enrique Pérez Herrero, Aug 22 2010 *)
f[p_, e_] := p^(2*e) + p^(2*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(n*sumdiv(n,d,moebius(d)^2/d^2),","))

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

Multiplicative with a(p^e) = p^(2*e) + p^(2*e-2). - Vladeta Jovovic, Dec 09 2001
a(n) = n^2 * Sum_{d|n} mu(d)^2/d^2 - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(d)^2*d^2. - Joerg Arndt, Jul 06 2011
Inverse Euler transform of n*A156733(n). - Paul D. Hanna and Vladeta Jovovic, Feb 14 2009
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_4(n)/(phi(n)*psi(n)) = A059377(n)/(A001615(n)*A000010(n))
a(n) = J_4(n)/J_2(n) = A059377(n)/A007434(n), where J_k is the k-th Jordan totient function. (End)
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s). Dirichlet convolution of A008966 and A000290. - R. J. Mathar, Apr 10 2011
G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^4 - 1)) = 1.5421162831401587416523241690601522041445615542162573163112157073779258386... - Vaclav Kotesovec, Sep 19 2020
a(n) = Sum_{d|n} d*phi(d)*psi(n/d). - Ridouane Oudra, Jan 01 2021
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} psi(gcd(n,k))*n/gcd(n,k), where psi(n) = A001615(n).
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 315*zeta(3)/Pi^6 = 0.393854... . - Amiram Eldar, Oct 19 2022

A065959 a(n) = n^3*Product_{distinct primes p dividing n} (1+1/p^3).

Original entry on oeis.org

1, 9, 28, 72, 126, 252, 344, 576, 756, 1134, 1332, 2016, 2198, 3096, 3528, 4608, 4914, 6804, 6860, 9072, 9632, 11988, 12168, 16128, 15750, 19782, 20412, 24768, 24390, 31752, 29792, 36864, 37296, 44226, 43344, 54432, 50654, 61740, 61544

Offset: 1

N. J. A. Sloane, Dec 08 2001

Enrique Pérez Herrero, Table of n, a(n) for n=1..10000
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
Wikipedia, Dedekind psi function.

Cf. A000010, A007434.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), this sequence (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

JordanTotient[n_,k_:1] := DivisorSum[n, #^k * MoebiusMu[n/#] &]/;(n>0) && IntegerQ[n]; A065959[n_] := JordanTotient[n,6] / JordanTotient[n,3]; Array[A065959, 39] (* Enrique Pérez Herrero, Aug 22 2010 *)
f[p_, e_] := p^(3*e) + p^(3*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(n^3*sumdiv(n,d,moebius(d)^2/d^3),","))

a(n)=sumdiv(n,d,moebius(n/d)^2*d^3); \\ Joerg Arndt, Jul 06 2011

Multiplicative with a(p^e) = p^(3*e)+p^(3*e-3). - Vladeta Jovovic, Dec 09 2001
a(n) = n^3*Sum_{d|n} mu(d)^2/d^3. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2*d^3. - Joerg Arndt, Jul 06 2011
a(n) = J_6(n)/J_3(n) = A069091(n)/A059376(n). - Enrique Pérez Herrero, Aug 22 2010
Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2*s). Dirichlet convolution of A008966 and A000578. - R. J. Mathar, Apr 10 2011
G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4. - Ilya Gutkovskiy, Oct 24 2018
From Vaclav Kotesovec, Sep 19 2020: (Start)
Sum_{k=1..n} a(k) ~ 105*n^4 / (4*Pi^4).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(p^6-1)) = 1.18370753651668075930203278269930233284040397061087910806697928843547863257... (End)

A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).

Original entry on oeis.org

1, 17, 82, 272, 626, 1394, 2402, 4352, 6642, 10642, 14642, 22304, 28562, 40834, 51332, 69632, 83522, 112914, 130322, 170272, 196964, 248914, 279842, 356864, 391250, 485554, 538002, 653344, 707282, 872644, 923522, 1114112, 1200644

Offset: 1

N. J. A. Sloane, Dec 08 2001

E. Pérez Herrero, Table of n, a(n) for n=1..10000
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
Wikipedia, Dedekind Psi function.

Cf. A000010, A007434, A059377, A069093.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), this sequence (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

A065960 := proc(n) n^4*mul(1+1/p^4,p=numtheory[factorset](n)) ; end proc:
seq(A065960(n),n=1..20) ; # R. J. Mathar, Jun 06 2011

a[n_] := n^4*DivisorSum[n, MoebiusMu[#]^2/#^4&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := p^(4*e) + p^(4*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(n^4*sumdiv(n,d,moebius(d)^2/d^4),","))

Multiplicative with a(p^e) = p^(4*e)+p^(4*e-4). - Vladeta Jovovic, Dec 09 2001
a(n) = n^4 * Sum_{d|n} mu(d)^2/d^4. - Benoit Cloitre, Apr 07 2002
a(n) = J_8(n)/J_4(n) = A069093(n)/A059377(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 29 2010
Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(2*s). - R. J. Mathar, Jun 06 2011
From Vaclav Kotesovec, Sep 19 2020: (Start)
Sum_{k=1..n} a(k) ~ 18711*zeta(5)*n^5 / Pi^10.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/(p^8-1)) = 1.078178802583045599985995264729541574821218371712364313741065126120993131... (End)

A351300 a(n) = n^5 * Product_{p|n, p prime} (1 + 1/p^5).

Original entry on oeis.org

1, 33, 244, 1056, 3126, 8052, 16808, 33792, 59292, 103158, 161052, 257664, 371294, 554664, 762744, 1081344, 1419858, 1956636, 2476100, 3301056, 4101152, 5314716, 6436344, 8245248, 9768750, 12252702, 14407956, 17749248, 20511150, 25170552, 28629152, 34603008, 39296688

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 5th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu).
Cf. A069095, A059378.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), this sequence (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(5*e) + p^(5*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^5);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^5 * mu(n/d)^2.
a(n) = n^5 * Sum_{d|n} mu(d)^2 / d^5.
Multiplicative with a(p^e) = p^(5*e) + p^(5*e-5). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^6 * zeta(6) / (6 * zeta(12)) = 225225 * n^6 / (1382 * Pi^6).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/(p^10-1)) = 1.03592823428850098309076014982275428113698561633329794485946580153004... (End)
a(n) = J_10(n) / J_5(n) = A069095(n) / A059378(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022

A351301 a(n) = n^6 * Product_{p|n, p prime} (1 + 1/p^6).

Original entry on oeis.org

1, 65, 730, 4160, 15626, 47450, 117650, 266240, 532170, 1015690, 1771562, 3036800, 4826810, 7647250, 11406980, 17039360, 24137570, 34591050, 47045882, 65004160, 85884500, 115151530, 148035890, 194355200, 244156250, 313742650, 387951930, 489424000, 594823322, 741453700

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 6th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu).

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), this sequence (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(6*e) + p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^6);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^6*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^6 * mu(n/d)^2.
a(n) = n^6 * Sum_{d|n} mu(d)^2 / d^6.
Multiplicative with a(p^e) = p^(6*e) + p^(6*e-6). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^7 * zeta(7) / (7 * zeta(14)) = 2606175 * n^7 * zeta(7) / (2 * Pi^14).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^6/(p^12-1)) = 1.01709659289559607702424749979498914920118274875188346777424441790304... (End)

A351303 a(n) = n^8 * Product_{p|n, p prime} (1 + 1/p^8).

Original entry on oeis.org

1, 257, 6562, 65792, 390626, 1686434, 5764802, 16842752, 43053282, 100390882, 214358882, 431727104, 815730722, 1481554114, 2563287812, 4311744512, 6975757442, 11064693474, 16983563042, 25700065792, 37828630724, 55090232674, 78310985282, 110522138624, 152588281250

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 8th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu).

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), this sequence (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(8*e) + p^(8*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^8);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^8*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^8 * mu(n/d)^2.
a(n) = n^8 * Sum_{d|n} mu(d)^2 / d^8.
Multiplicative with a(p^e) = p^(8*e) + p^(8*e-8). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^9 * zeta(9) / (9 * zeta(18)) = 4331032831125 * n^9 * zeta(9) / (43867 * Pi^18).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^8/(p^16-1)) = 1.004062071480173688638170669970682370243496458304179434830922739661777... (End)
a(n) = J_16(n)/J_8(n) = J_16(n)/A069093(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 14 2022

A351304 a(n) = n^9 * Product_{p|n, p prime} (1 + 1/p^9).

Original entry on oeis.org

1, 513, 19684, 262656, 1953126, 10097892, 40353608, 134479872, 387440172, 1001953638, 2357947692, 5170120704, 10604499374, 20701400904, 38445332184, 68853694464, 118587876498, 198756808236, 322687697780, 513000262656, 794320419872, 1209627165996, 1801152661464, 2647101800448

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 9th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu).

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (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), this sequence (k=9), A351305 (k=10).

f[p_, e_] := p^(9*e) + p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^9);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^9*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

from math import prod
from sympy import factorint
def A351304(n): return prod(p**(9*e)+p**(9*(e-1)) for p,e in factorint(n).items()) # Chai Wah Wu, Sep 28 2024

a(n) = Sum_{d|n} d^9 * mu(n/d)^2.
a(n) = n^9 * Sum_{d|n} mu(d)^2 / d^9.
Multiplicative with a(p^e) = p^(9*e) + p^(9*e-9). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^10 * zeta(10) / (10 * zeta(20)) = 3273645375 * n^10 / (349222 * Pi^10).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^9/(p^18-1)) = 1.002004575331916689985388864168116922608947780516939765639888137700557... (End)

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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A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

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A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).

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A065959 a(n) = n^3*Product_{distinct primes p dividing n} (1+1/p^3).

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A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).

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