A306016 -id:A306016 - cp's OEIS frontend

A007427 Moebius transform applied twice to sequence 1,0,0,0,....

1, -2, -2, 1, -2, 4, -2, 0, 1, 4, -2, -2, -2, 4, 4, 0, -2, -2, -2, -2, 4, 4, -2, 0, 1, 4, 0, -2, -2, -8, -2, 0, 4, 4, 4, 1, -2, 4, 4, 0, -2, -8, -2, -2, -2, 4, -2, 0, 1, -2, 4, -2, -2, 0, 4, 0, 4, 4, -2, 4, -2, 4, -2, 0, 4, -8, -2, -2, 4, -8, -2, 0, -2, 4, -2, -2, 4, -8, -2, 0, 0

Offset: 1

N. J. A. Sloane

|a(n)| is the number of ways to write n as a product of 2 squarefree numbers (i.e., number of ways to write n = x*y with 1 <= x <= n, 1 <= y <= n, x and y squarefree). - Benoit Cloitre, Jan 01 2003

			G.f. = x - 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 4*x^6 - 2*x^7 + x^9 + 4*x^10 + ...
We have a(3^1) = C(2, 1)*(-1)^1 = -2, a(3^2) = C(2, 2)*(-1)^2 = 1, and a(3^m) = C(2, m)*(-1)^m = 0 for m >= 3. - _Petros Hadjicostas_, Jun 07 2019

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
N. J. A. Sloane, Transforms.
Wikipedia, Adolf Piltz.

Dirichlet inverse of A000005, Mobius transform of A008683.
Cf. A063524, A007428, A005117, A008836, A046099, A225817.
Cf. A013661, A001620, A306016.

a007427 n = sum $ zipWith (*) mds $ reverse mds where
   mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013

möbius := proc(a)  local b, i, mo: b := NULL:
mo := (m,n) -> `if`(irem(m,n) = 0, numtheory:-mobius(m/n), 0);
for i to nops(a) do b := b, add(mo(i,j)*a[j], j=1..i) od: [b] end:
(möbius@@2)([1, seq(0, i=1..80)]); # Peter Luschny, Sep 08 2017

f[n_] := Plus @@ Times @@@ (MoebiusMu[{#, n/#}] & /@ Divisors@n); Array[f, 105] (* Robert G. Wilson v *)
a[n_] := DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, direuler(p=2, n, (1 - X)^2)[n])}; /* Michael Somos, Nov 15 2002 */

{a(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d)))}; /* Michael Somos, Nov 15 2002 */

a(n)=if(n>1,my(f=factor(n)[,2],s=sum(i=1,#f,f[i]==1));if(vecmax(f)>2,0,(-1)^s<Charles R Greathouse IV, Jun 28 2011

from math import prod, comb
from sympy import factorint
def A007427(n): return prod(-comb(2,e) if e&1 else comb(2,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

Dirichlet g.f.: 1/zeta(s)^2.
Multiplicative function with a(p^e) = binomial(2, e)*(-1)^e for p prime and e >= 0.
a(n) = Sum_{d|n} mu(d)*mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n^2) = A008683(n)^2. a(A005117(n)) = (-2)^A001221(A005117(n)). - Enrique Pérez Herrero, Jun 27 2011 [Misrendering of contribution rectified by Peter Munn, Mar 06 2020]
a(n) is the Dirichlet inverse of A000005, which means a(n) = -Sum_{d|n, dA000005(n/d)*a(d). - Enrique Pérez Herrero, Jan 19 2013
a(n) = 0 if n is not cubefree: A046099, otherwise sign(a(n)) = lambda(n), where lambda is A008836. - Enrique Pérez Herrero, Jan 19 2013
Dirichlet g.f. of |a(n)|: zeta(s)^2/zeta(2s)^2 (conjectured). - Ralf Stephan, Jul 05 2013.  The conjecture is correct because 1+Sum_{e>=1} binomial(2,e)/p^(e*s) = (p^s+1)^2/p^2s, whose product over p is zeta(s)^2/zeta(2s)^2. - Michael Shamos
a(n) = Sum_{k=1..A000005(n)} A225817(n,k)*A225817(n,n+1-k). - Reinhard Zumkeller, Jul 30 2013
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} tau(k)*A(x^k), where tau = A000005. - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ (n/zeta(2)^2) * (log(n) + 2*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 24 2023

Added a proof of Stephan's conjecture about the Dirichlet g.f. of |a(n)|.

A048105 Number of non-unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 2, 2, 5, 0, 0, 0, 4, 0

Offset: 1

Labos Elemer

nonn

Number of zeros in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013

			Example 1: If n is squarefree (A005117) then a(n)=0 since all divisors are unitary.
Example 2: n=12, d(n)=6, ud(n)=4, nud(12)=d(12)-ud(12)=2; from {1,2,3,4,6,12} {1,3,4,12} are unitary while {2,6} are not unitary divisors.
Example 3: n=p^k, a true prime power, d(n)=k+1, u(d)=2^r(x)=2, so nud(n)=d(p^k)-2=k+1 i.e., it can be arbitrarily large.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A007875, A056170, A225817, A034444.

Cf. A001620, A229099, A306016.

a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]
-- Reinhard Zumkeller, Aug 17 2011

with(NumberTheory):
seq(SumOfDivisors(n, 0) - 2^NumberOfPrimeFactors(n, 'distinct'), n = 1..105);
# Peter Luschny, Jul 27 2023

Table[DivisorSigma[0, n] - 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Dec 10 2014 *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)-2^#f \\ Charles R Greathouse IV, Sep 18 2015

from math import prod
from sympy import factorint
def A048105(n): return -(1<Chai Wah Wu, Aug 12 2024

a(n) = Sigma(0, n) - 2^r(n), where r() = A001221, the number of distinct primes dividing n.
From Reinhard Zumkeller, Jul 30 2013: (Start)
a(n) = A000005(n) - A034444(n).
For n > 1: a(n) = A000005(n) - 2 * A007875(n). (End)
Dirichlet g.f.: zeta(s)^2 - zeta(s)^2/zeta(2*s). - Geoffrey Critzer, Dec 10 2014
G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 21 2017
Sum_{k=1..n} a(k) ~ (1-6/Pi^2)*n*log(n) + ((1-6/Pi^2)*(2*gamma-1)+(72*zeta'(2)/Pi^4))*n , where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

A060594 Number of solutions to x^2 == 1 (mod n), that is, square roots of unity modulo n.

Original entry on oeis.org

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

Offset: 1

Jud McCranie, Apr 11 2001

Sum_{k=1..n} a(k) appears to be asymptotic to C*n*log(n) with C = 0.6... - Benoit Cloitre, Aug 19 2002
a(q) is the number of real characters modulo q. - Benoit Cloitre, Feb 02 2003
Also number of real Dirichlet characters modulo n and Sum_{k=1..n}a(k) is asymptotic to (6/Pi^2)*n*log(n). - Steven Finch, Feb 16 2006
Let P(n) be the product of the numbers less than and coprime to n. By theorem 59 in Nagell (which is Gauss's generalization of Wilson's theorem): for n > 2, P == (-1)^(a(n)/2) (mod n). - T. D. Noe, May 22 2009
Shadow transform of A005563. - Michel Marcus, Jun 06 2013
For n > 2, a(n) = 2 iff n is in A033948. - Max Alekseyev, Jan 07 2015
For n > 1, number of square numbers on the main diagonal of an (n-1) X (n-1) square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 19 2021
a(n) is the number of linear Alexander quandles (Z/nZ, k) that are kei (equivalently, that have good involutions); see Ta, Prop. 5.6 and cf. A387317. - Luc Ta, Aug 26 2025

			The four numbers 1^2, 3^2, 5^2 and 7^2 are congruent to 1 modulo 8, so a(8) = 4.

Trygve Nagell, Introduction to Number Theory, AMS Chelsea, 1981, p. 100. [From T. D. Noe, May 22 2009]
Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisé, 1995, Collection SMF, p. 260.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, pp. 196-197.

T. D. Noe, Table of n, a(n) for n = 1..1000
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Keith Matthews, Solving the congruence x^2=a(mod m).
Emilia Mezzetti and Rosa Maria Miró-Roig, Togliatti systems and Galois coverings, arXiv preprint arXiv:1611.05620 [math.AG], 2016-2018. See Lemma 6.1.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4].
N. J. A. Sloane, Transforms.
Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A000010, A005087, A033948, A046073, A073103 (x^4 == 1 (mod n)).
Cf. A001620, A306016.
Cf. A387317.

A060594 := proc(n)
   option remember;
   local a,b,c;
   if type(n,even) then
     a:= padic:-ordp(n,2);
     b:= 2^a;
     c:= n/b;
     min(b/2, 4) * procname(c)
   else
     2^nops(numtheory:-factorset(n))
   fi
end proc:
map(A060594, [$1 .. 100]); # Robert Israel, Jan 05 2015

a[n_] := Sum[ Boole[ Mod[k^2 , n] == 1], {k, 1, n}]; a[1] = 1; Table[a[n], {n, 1, 103}] (* Jean-François Alcover, Oct 21 2011 *)
a[n_] := Switch[Mod[n, 8], 2|6, 2^(PrimeNu[n]-1), 1|3|4|5|7, 2^PrimeNu[n], 0, 2^(PrimeNu[n]+1)]; Array[a, 103] (* Jean-François Alcover, Apr 09 2016 *)

PARI
```
a(n)=sum(i=1,n,if((i^2-1)%n,0,1))
```

a(n)=my(o=valuation(n,2));2^(omega(n>>o)+max(min(o-1,2),0)) \\ Charles R Greathouse IV, Jun 06 2013

a(n)=if(n<=2, 1, 2^#znstar(n)[3] ); \\ Joerg Arndt, Jan 06 2015

from sympy import primefactors
def a007814(n): return 1 + bin(n - 1).count('1') - bin(n).count('1')
def a(n):
    if n%2==0:
        A=a007814(n)
        b=2**A
        c=n//b
        return min(b//2, 4)*a(c)
    else: return 2**len(primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 18 2017, after the Maple program

from sympy import primefactors
def A060594(n): return (1<>(s:=(~n & n-1).bit_length()))))*(1 if n&1 else 1<Chai Wah Wu, Oct 26 2022

print([len(Integers(n).square_roots_of_one()) for n in range(1,100)]) # Ralf Stephan, Mar 30 2014

If n == 0 (mod 8), a(n) = 2^(A005087(n) + 2); if n == 4 (mod 8), a(n) = 2^(A005087(n) + 1); otherwise a(n) = 2^(A005087(n)). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
a(n) = 2^omega(n)/2 if n == +/-2 (mod 8), a(n) = 2^omega(n) if n== +/-1, +/-3, 4 (mod 8), a(n) = 2*2^omega(n) if n == 0 (mod 8), where omega(n) = A001221(n). - Benoit Cloitre, Feb 02 2003
For n >= 2 A046073(n) * a(n) = A000010(n) = phi(n). This gives a formula for A046073(n) using the one in A060594(n). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
Multiplicative with a(2) = 1; a(2^2) = 2; a(2^e) = 4 for e > 2; a(q^e) = 2 for q an odd prime. - Eric M. Schmidt, Jul 09 2013
a(n) = 2^A046072(n) for n>2, in accordance with the above formulas by Ahmed Fares. - Geoffrey Critzer, Jan 05 2015
a(n) = Sum_{k=1..n} floor(sqrt(1+n*(k-1)))-floor(sqrt(n*(k-1))). - Wesley Ivan Hurt, May 19 2021
From Amiram Eldar, Dec 30 2022: (Start)
Dirichlet g.f.: (1-1/2^s+2/4^s)*zeta(s)^2/zeta(2*s).
Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*gamma - 1 - log(2)/2 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)

A018892 Number of ways to write 1/n as a sum of exactly 2 unit fractions.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 14, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 14, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 23, 2, 5, 8, 7, 5, 14, 2, 8, 5, 14, 2, 18, 2, 5, 8, 8, 5, 14, 2, 14, 5, 5, 2, 23, 5, 5, 5, 11, 2, 23, 5, 8, 5, 5, 5, 17, 2, 8, 8

Offset: 1

Robert G. Wilson v

Number of elements in the set {(x,y): x|n, y|n, x<=y, gcd(x,y)=1}. Number of divisors of n^2 less than or equal to n. - Vladeta Jovovic, May 03 2002
Equivalently, number of pairs (x,y) such that lcm(x,y)=n. - Benoit Cloitre, May 16 2002
Also, number of right triangles with an integer hypotenuse and height n. - Reinhard Zumkeller, Jul 10 2002
The triangles are to be considered as resting on their hypotenuse, with the height measured to the right angle. - Franklin T. Adams-Watters, Feb 19 2015
a(n) >= 2 for n>=2 because of the identities 1/n = 1/(2*n) + 1/(2*n) = 1/(n+1) + 1/(n*(n+1)). - Lekraj Beedassy, May 04 2004
a(n) is the number of divisors of n^2 that are <= n; e.g., a(12) counts these 8 divisors of 12: 1,2,3,4,6,8,9,12. - Clark Kimberling, Apr 21 2019

			n=1: 1/1 = 1/2 + 1/2.
n=2: 1/2 = 1/4 + 1/4 = 1/3 + 1/6.
n=3: 1/3 = 1/6 + 1/6 = 1/4 + 1/12.

K. S. Brown, Posting to netnews group sci.math, Aug 17 1996.
L. E. Dickson, History of The Theory of Numbers, Vol. 2 p. 690, Chelsea NY 1923.
A. M. and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1, Dover, N.Y., 1987, pp. 8 and 60, Problem 19.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892.
Roger B. Eggleton, Problem 10501(a), American Mathematical Monthly, Vol. 105, No. 4, (1998), p. 372.
Project Euler, Problem 379: Least common multiple count.

Records: A126097, A126098.
Cf. A000010, A001221, A046079, A048691, A063647, A089233, A184389.
Cf. A001620, A082633, A013661, A201994, A306016.

a018892 n = length [d | d <- [1..n], n^2 `mod` d == 0]
-- Reinhard Zumkeller, Jan 08 2012

f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; Table[f[2, n], {n, 96}] (* Robert G. Wilson v, Aug 03 2005 *)
a[n_] := (DivisorSigma[0, n^2] + 1)/2; Table[a[n], {n, 1, 99}](* Jean-François Alcover, Dec 19 2011, after Vladeta Jovovic *)

A018892(n)=(numdiv(n^2)+1)/2 \\ M. F. Hasler, Dec 30 2007

A018892s(n)=local(t=divisors(n^2));vector((#t+1)/2,i,[n+t[i],n+n^2/t[i]]) /* show solutions */ \\ M. F. Hasler, Dec 30 2007

a(n)=sumdiv(n,d,sum(i=1,d,lcm(d,i)==n)) \\ Charles R Greathouse IV, Apr 08 2012

from math import prod
from sympy import factorint
def A018892(n): return prod((a<<1)+1 for a in factorint(n).values())+1>>1 # Chai Wah Wu, Aug 20 2023

If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1 + 1)(2*a2 + 1) ... (2*at + 1) + 1)/2.
a(n) = (tau(n^2)+1)/2. - Vladeta Jovovic, May 03 2002
a(n) = A063647(n)+1 = A046079(2*n)+1. - Lekraj Beedassy, Dec 01 2003
a(n) = Sum_{d|n} phi(2^omega(d)), where phi is A000010 and omega is A001221. - Enrique Pérez Herrero, Apr 13 2012
a(n) = A000005(n) + A089233(n). - James Spahlinger, Feb 16 2016
a(n) = n + Sum_{i=1..n} sign(n^2 mod -i). - Wesley Ivan Hurt, Apr 07 2021
a(n) = Sum_{d|n} mu(n/d)*A184389(d). - Ridouane Oudra, Feb 22 2022
Sum_{k=1..n} a(k) ~ (n/(2*zeta(2)))*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 + zeta(2) + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Oct 03 2024

More terms from David W. Wilson, Sep 15 1996
First example corrected by Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 02 2009
Incorrect Mathematica program deleted by N. J. A. Sloane, Jul 08 2009

A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 13, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 13, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 22, 1, 4, 7, 6, 4, 13, 1, 7, 4, 13, 1, 17, 1, 4, 7, 7, 4, 13, 1, 13, 4, 4, 1, 22, 4, 4, 4, 10, 1, 22, 4, 7, 4, 4, 4

Offset: 1

Henry Bottomley, Jul 23 2001

Also number of ways to write 1/n as sum of exactly two distinct unit fractions. - Thomas L. York, Jan 11 2014
Also number of positive integers m such that 1/n + 1/m is a unit fraction. - Jon E. Schoenfield, Apr 17 2018
If 1/n = 1/b - 1/c then n = bc/(c-b) and 1/n = 1/(2n-b) + 1/(c+2n) (though it is also the case that 1/n = 1/(2n) + 1/(2n) equivalent to b = c = 0).
Also number of divisors of n^2 less than n. - Vladeta Jovovic, Aug 13 2001
Number of elements in the set {(x,y): x|n, y|n, xVladeta Jovovic, May 03 2002
Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre, Jan 04 2002
This is similar to A062799, having the same first 29 terms. But they are different sequences.
If A001221(n) = omega(n) <= 2, then a(n) = A062799(n); if A001221(n) > 2, then a(n) > A062799(n). - Matthew Vandermast, Aug 25 2004
Number of r X s integer-sided rectangles such that r + s = 4n, r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020
Also number of integer-sided right triangles with 2n as a leg. Equivalent to the even indices of A046079. - Nathaniel C Beckman, May 14 2020; Jun 26 2020
a(n) is the number of positive integers k such that k+n divides k*n. - Thomas Ordowski, Dec 02 2024

			a(10) = 4 since 1/10 = 1/5 - 1/10 = 1/6 - 1/15 = 1/8 - 1/40 = 1/9 - 1/90.
a(12) = 7: the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the decompositions are (1, 2), (1, 3), (1, 4), (1, 6), (1, 12), (2, 3), (3, 4).

T. D. Noe, Table of n, a(n) for n = 1..10000
Christopher J. Bradley, Solution to Problem 2175, Crux Mathematicorum, Vol. 23, No. 7, (Nov 1997), pp. 443-444.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Roger B. Eggleton, Unitary Fractions: 10501, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 372.
Amiram Eldar, Plot of (Sum_{i=1..n} a(i))/f(n) - 1, where f(n) is an asymptotic function, for n = 2^(1..28).
Amiram Eldar, Plot of Sum_{i=1..n} a(i)/(n*log(n)*log(log(n))) for n = 2^(1..28).
Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001, pp. 303-306.

Cf. A018892, A063427, A063428, A129296.
First twenty-nine terms identical to those of A062799.
Cf. A001221, A046079, A048691, A063717, A063718.
Cf. A001620, A082633, A013661, A201994, A306016.

[(NumberOfDivisors(n^2)-1)/2 : n in [1..100]]; // Vincenzo Librandi, Apr 18 2018

Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
(DivisorSigma[0,Range[100]^2]-1)/2 (* Harvey P. Dale, Apr 15 2013 *)

for(n=1,100,print1(sum(i=1,n^2,if((n*i)%(i+n),0,1)),","))

a(n)=numdiv(n^2)\2 \\ Charles R Greathouse IV, Oct 03 2016

a(n) = (tau(n^2)-1)/2.
a(n) = A018892(n)-1. If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1+1)(2*a2+1)...(2*at+1)-1)/2.
If n is prime a(n)=1. Conjecture: (1/n)*Sum_{i=1..n} a(i) = C*log(n)*log(log(n)) + o(log(n)) with C=0.7... [The conjecture is false. See the plot and the asymptotic formula below. - Amiram Eldar, Oct 03 2024]
Bisection of A046079. - Lekraj Beedassy, Jul 09 2004
a(n) = Sum_{i=1..2*n-1} (1 - ceiling(i*(4*n-i)/(4*n-2*i)) + floor(i*(4*n-i)/(4*n-2*i))). - Wesley Ivan Hurt, Apr 24 2020
Sum_{k=1..n} a(k) ~ (n/(2*zeta(2)))*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 - zeta(2) + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Oct 03 2024

A327527 Number of uniform divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 7, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4, 8, 2, 5, 5, 7, 2, 8, 2, 6, 8

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

			The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

See link for additional cross-references.
Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.
Cf. A072774, A327526.
Cf. A001620, A013661, A306016, A368250, A368251.

Table[Length[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Dec 19 2023 *)

isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
A327527(n) = sumdiv(n,d,isA072774(d)); \\ Antti Karttunen, Nov 13 2021

From Amiram Eldar, Dec 19 2023: (Start)
a(n) = A034444(n) + A368251(n).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2) + c * zeta(2)), where gamma is Euler's constant (A001620) and c = A368250. (End)

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

Original entry on oeis.org

1, 3, 2, 4, 2, 6, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 2, 6, 2, 8, 4, 6, 2, 8, 2, 6, 2, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 8, 4, 6, 2, 8, 2, 6, 4, 8, 2, 6, 4, 8, 4, 6, 2, 16, 2, 6, 4, 4, 4, 12, 2, 8, 4, 12, 2, 8, 2, 6, 4, 8, 4, 12, 2, 8, 2, 6, 2, 16, 4

Offset: 1

Andrey Zabolotskiy, May 07 2018

			There are 6 = A001615(4) lattices in Z^2 whose quotient group is C_4. The reflection through an axis relates <(4,0), (1,1)> and <(4,0), (3,1)>. The remaining 4 = a(4) lattices are fixed.

Álvar Ibeas, Table of n, a(n) for n = 1..10000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4. Contains errors for n = 24 and 28.]

Cf. A069735 (not only primitive sublattices), A304183 (primitive oblique sublattices), A069734 (all sublattices).
Cf. other columns of tables 4 and 5 from [Rutherford, 2009]: A001615, A060594, A157223, A000089, A157224, A000086, A157227, A019590, A157228, A157226, A157230, A157231, A154272, A157235.
Cf. A034444, A001221, A001620, A306016.

f[p_, e_] := If[p == 2, If[e == 1, 3, 4], 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

From Álvar Ibeas, Mar 18 2021: (Start)
For n odd, a(n) = A034444(n) = 2^(A001221(n)).
For n even, a(n) = A034444(n) + A034444(n/2). If 4|n, a(n) = 2^(A001221(n) + 1); otherwise, a(n) = 3 * 2^(A001221(n) - 1).
Multiplicative with a(2) = 3, a(2^e) = 4 (for e>1), and a(p^e) = 2 (for p>2).
Dirichlet g.f.: (1+2^(-s)) * zeta(s)^2 / zeta(2s).
(End)
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*9*n/Pi^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 31 2022

A055205 Number of nonsquare divisors of n^2.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 9, 5, 5, 1, 13, 2, 5, 3, 9, 1, 19, 1, 5, 5, 5, 5, 16, 1, 5, 5, 13, 1, 19, 1, 9, 9, 5, 1, 17, 2, 9, 5, 9, 1, 13, 5, 13, 5, 5, 1, 33, 1, 5, 9, 6, 5, 19, 1, 9, 5, 19, 1, 23, 1, 5, 9, 9, 5, 19, 1, 17, 4, 5, 1, 33, 5, 5, 5, 13, 1, 33, 5, 9, 5, 5, 5

Offset: 1

Labos Elemer, Jun 19 2000

Seems to be equal to the number of unordered pairs of coprime divisors of n. (Checked up to 2*10^14.) - Charles R Greathouse IV, May 03 2013
Outline of a proof for this observation, R. J. Mathar, May 05 2013: (Start)
i) To construct the divisors of n, write n=product_i p_i^e_i as the standard prime power decomposition, take any subset of the primes p_i (including the empty set representing the 1) and run with the associated list exponents from 0 up to their individual e_i.
To construct the *nonsquare* divisors of n, ensure that one or more of the associated exponents is/are odd. (The empty set is interpreted as 1^0 with even exponent.) To construct the nonsquare divisors of n^2, the principle remains the same, although the exponents may individually range from 0 up to 2*e_i.
The nonsquare divisor is therefore a nonempty product of prime powers (at least one) with odd exponents times  a (potentially empty) product of prime powers (of different primes) with even exponents.
The nonsquare divisors of n^2 have exponents from 0 up to 2*e_i, but the subset of exponents in the "even/square" factor has e_i candidates (range 2, 4, .., 2*e_i) and in the "odd/nonsquare" factor also only e_i candidates (range 1,3,5,2*e_i-1).
ii) To construct the pairs of coprime divisors of n, take any two non-intersecting subsets of  the set of p_i (possibly the empty subset which represents the factor 1), and let the exponents run from 1 up to their individual e_i in each of the two products.
iii) The bijection between the sets constructed in i) and ii) is given by mapping the two non-intersection prime sets onto each other, and observing that the numbers of compositions of exponents have the same orders in both cases.
(End)

			n = 8, d(64) = 7 and from the 7 divisors {1,4,16,64} are square and the remaining 3 = a(8).
n = 12, d(144) = 15, from which 6 divisors are squares {1,4,9,16,36,144} so a(12) = d(144)-d(12) = 9
a(60) = (number of terms of finite A171425) = 33. [_Reinhard Zumkeller_, Dec 08 2009]

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000005, A000290, A048691, A056595.

Cf. A001620, A013661, A082633, A201994, A306016.

a055205 n = length [d | d <- [1..n^2], n^2 `mod` d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011

Table[Count[Divisors[n^2], d_ /;  ! IntegerQ[Sqrt[d]]], {n, 1, 95}] (* Jean-François Alcover, Mar 22 2011 *)
Table[DivisorSigma[0,n^2]-DivisorSigma[0,n],{n,100}] (* Harvey P. Dale, Sep 02 2017 *)

a(n)=my(f=factor(n)[,2]);prod(i=1,#f,2*f[i]+1)-prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, May 02 2013

a(n) = A000005(n^2)-A000005(n) because the number of square divisors of n^2 equals the number of divisors of n.
a(n) = A056595(A000290(n)).
a(n) = A048691(n) - A000005(n). - Reinhard Zumkeller, Dec 08 2009
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n)^2/2 + c_1 * log(n) + c_2), where c_1 =  3*gamma - 2*zeta'(2)/zeta(2) - zeta(2) - 1 = 0.226634..., c_2 = 3*gamma^2 - (2*gamma - 1)*zeta(2) - 3*gamma_1 + (1 - 3*gamma)*(2*zeta'(2)/zeta(2) + 1) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2) = -0.0529271..., gamma is Euler's constant (A001620), and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Dec 01 2023

A383156 The sum of the maximum exponents in the prime factorizations of the divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 7, 1, 3, 3, 10, 1, 7, 1, 7, 3, 3, 1, 13, 3, 3, 6, 7, 1, 7, 1, 15, 3, 3, 3, 13, 1, 3, 3, 13, 1, 7, 1, 7, 7, 3, 1, 21, 3, 7, 3, 7, 1, 13, 3, 13, 3, 3, 1, 15, 1, 3, 7, 21, 3, 7, 1, 7, 3, 7, 1, 22, 1, 3, 7, 7, 3, 7, 1, 21, 10, 3, 1

Offset: 1

Amiram Eldar, Apr 18 2025

Inverse Möbius transform of A051903.

a(n) depends only on the prime signature of n (A118914).

			4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = 0 + 1 + 2 = 3.
12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = 0 + 1 + 1 + 2 + 1 + 2 = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of Sum_{k=1..n} a(k)/(c_1*n*log(n) + c_2*n) - 1 for n = 10^(1..10).

Cf. A000005, A001221, A005117, A034444, A051903, A073184, A118914, A252505, A309307, A383157, A383158, A383159.

Cf. A001620, A013661, A033150, A306016.

emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := DivisorSum[n, emax[#] &]; Array[a, 100]
(* second program: *)
a[n_] := If[n == 1, 0, Module[{e = FactorInteger[n][[;; , 2]], emax, v}, emax = Max[e]; v = Table[Times @@ (Min[# + 1, k + 1] & /@ e), {k, 1, emax}]; v[[1]] + Sum[k*(v[[k]] - v[[k - 1]]), {k, 2, emax}] - 1]]; Array[a, 100]

emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sumdiv(n, d, emax(d));

a(n) = if(n == 1, 0, my(e = factor(n)[, 2], emax = vecmax(e), v); v = vector(emax, k, vecprod(apply(x ->min(x+1 , k+1), e))); v[1] + sum(k = 2, emax, k * (v[k]-v[k-1])) - 1);

a(n) = Sum_{d|n} A051903(d).
a(n) = A000005(n) * A383157(n)/A383158(n).
a(p^e) = e*(e+1)/2 for prime p and e >= 0. In particular, a(p) = 1 for prime p.
a(n) = 2^omega(n) - 1 = A309307(n) if n is squarefree (A005117).
a(n) = tau(n, 2) - 1 + Sum_{k=2..A051903(n)} k * (tau(n, k+1) - tau(n, k)), where tau(n, k) is the number of k-free divisors of n (k-free numbers are numbers that are not divisible by a k-th power other than 1). For a given k >= 2, tau(n, k) is a multiplicative function with tau(p^e, k) = min(e+1, k). E.g., tau(n, 2) = A034444(n), tau(n, 3) = A073184(n), and tau(n, 4) = A252505(n).
Sum_{k=1..n} a(k) ~ c_1 * n * log(n) + c_2 * n, where c_1 is Niven's constant (A033150), c_2 = -1 + (2*gamma - 1)*c_1 - 2*zeta'(2)/zeta(2)^2 - Sum_{k>=3} (k-1) * (k*zeta'(k)/zeta(k)^2 - (k-1)*zeta'(k-1)/zeta(k-1)^2) = -2.37613633493572231366..., and gamma is Euler's constant (A001620).

A124315 a(n) = Sum_{ d divides n } tau(gcd(d,n/d)), where tau = sigma_0 = A000005.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 9, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 12, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 18, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 16, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 18, 9, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 24, 2, 8

Offset: 1

Robert G. Wilson v, Sep 30 2006

Apparently the Mobius transform of A046951. - R. J. Mathar, Feb 07 2011

Number of ordered pairs of divisors of n, (d1,d2), with d1<=d2, such that d1|d2 and n|(d1*d2). - Wesley Ivan Hurt, Mar 22 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000005, A001616, A001620, A046951, A061884, A124316, A306016.

A124315 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do igcd(d,n/d) ; a := a+numtheory[tau](%) ; end do: a; end proc: # R. J. Mathar, Apr 14 2011

Table[Plus @@ Map[DivisorSigma[0, GCD[ #, n/# ]] &, Divisors@n], {n, 98}]
f[p_, e_] := e + 1 + Floor[e^2/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

a(n) = sumdiv(n, d, numdiv(gcd(d, n/d))); \\ Michel Marcus, Feb 12 2016

from sympy import divisors, divisor_count, gcd
def a(n): return sum([divisor_count(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

a(p) = 2 iff p is a prime.
Multiplicative with a(p^e) = e+1+floor(e^2/4). - R. J. Mathar, Apr 14 2011
Dirichlet g.f.: zeta(s)^2 * zeta(2*s). - Vaclav Kotesovec, Jan 11 2019
Sum_{k=1..n} a(k) ~ (Pi^2/6) * (n*log(n) + (2*gamma - 1 + 2*zeta'(2)/zeta(2))*n), where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 22 2022

cp's OEIS Frontend

A007427 Moebius transform applied twice to sequence 1,0,0,0,....

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A048105 Number of non-unitary divisors of n.

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A060594 Number of solutions to x^2 == 1 (mod n), that is, square roots of unity modulo n.

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A018892 Number of ways to write 1/n as a sum of exactly 2 unit fractions.

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A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

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