A048691 -id:A048691 - cp's OEIS frontend

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

First differs from A066620 at a(210) = 36, A066620(210) = 35.

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {2,3}    {2,3}  {2,3}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,5}    {2,9}  {2,5}    {2,9}  {3,4}
                {3,8}  {3,5}    {3,4}  {3,4}    {3,4}  {3,8}
                       {5,6}    {4,9}  {3,5}    {3,8}  {3,16}
                       {2,15}          {4,5}    {4,9}  {3,32}
                       {3,10}          {5,6}    {8,9}
                       {2,3,5}         {2,15}
                                       {3,10}
                                       {3,20}
                                       {4,15}
                                       {5,12}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 39, 56, 77, 103, 134, 174, 223, 283, 356, 445, 547, 666, 802, 959, 1139, 1344, 1574, 1835, 2128, 2454, 2815, 3213, 3648, 4126, 4653, 5239, 5888, 6608, 7407, 8298, 9288, 10385, 11597, 12936, 14408, 16025, 17799, 19746, 21882, 24221

Offset: 1

Gus Wiseman, Apr 29 2021

nonn

			The a(7) = 27 divisors:
  1  32  81  64  25  6  1
     16  27  32  5   3
     8   9   16  1   2
     4   3   8       1
     2   1   4
     1       2
             1

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Antidiagonal row sums (row sums of the triangle) of A343656.
Dominated by A343661.
A000005(n) counts divisors of n.
A000312(n) = n^n.
A007318(n,k) counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481(n,k) counts k-multisets of elements of {1..n}.
A343658(n,k) counts k-multisets of divisors of n.
Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.

Total/@Table[DivisorSigma[0,k^(n-k)],{n,30},{k,n}]

a(n) = Sum_{k=1..n} A000005(k^(n-k)).

A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

Original entry on oeis.org

1, 3, 3, 0, 3, 9, 3, 0, 0, 9, 3, 0, 3, 9, 9, 0, 3, 0, 3, 0, 9, 9, 3, 0, 0, 9, 0, 0, 3, 27, 3, 0, 9, 9, 9, 0, 3, 9, 9, 0, 3, 27, 3, 0, 0, 9, 3, 0, 0, 0, 9, 0, 3, 0, 9, 0, 9, 9, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 9, 27, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 0, 9, 3, 0, 9, 9, 9, 0, 3, 0, 9, 0, 9, 9, 9, 0, 3, 0, 0, 0

Offset: 1

Vaclav Kotesovec, Aug 20 2021

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A008966, A074823, A165824.
Cf. A013928, A069201, A048691.
Cf. A001620, A082633.

Table[MoebiusMu[n]^2 * 3^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^3)[n], ", "))

Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)), then Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where f(1) = Product_{primes p} (1 - 6/p^2 + 8/p^3 - 3/p^4) = 0.1148840440802287887292512767015990978487135526872830176248484270625666728..., f'(1) = f(1) * Sum_{primes p} 12*log(p) / ((p-1)*(p+3)) = 0.5497153490016133577871571904347511299324572220423331992393596243955677299..., f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-24*p*(p-1) * log(p)^2 / ((p-1)^2 * (p+3)^2)) = 0.9028322988288094236586622799305270026576436536391185119652318723470259904... and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633).
a(n) = A008966(n) * A048691(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 3, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 11, 11, 18, 12, 24, 14, 24, 24, 20, 18, 33, 20, 36, 32, 36, 24, 44, 27, 42, 30, 48, 30, 72, 32, 37, 48, 54, 48, 66, 38, 60, 56, 66, 42, 96, 44, 72, 66, 72, 48, 80, 51, 81, 72, 84, 54, 90, 72, 88, 80, 90, 60, 144, 62, 96, 88, 70, 84, 144, 68

Offset: 1

Amiram Eldar, Feb 10 2023

a(n) is the sum of delta_d(n) over the divisors d of n, where delta_d(n) is the greatest divisor of n that is relatively prime to n.
Denoted by Sur(n) in Khan (2005).
Related sequences: A048691(n) = Sum_{d|n} #{d'; d' | n, gcd(d, d') = 1}, and A328485(n) = Sum_{d|n} Sum_{d' | n, gcd(d, d') = 1} d' (number and sum of divisors instead of maximal divisor, respectively).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mizan R. Khan, Problem 10922, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, The Sum of Divisors Won't Die, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Cf. A000203, A005117, A034448, A360523.
Cf. A065465, A072691, A152649, A330523.
Cf. A048691, A328485.

f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]);}

Multiplicative with a(p^e) = p^e + e.
Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).
liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).
limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).

A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

Original entry on oeis.org

0, 3, 10, 16, 37, 10, 42, 24, 58, 53, 164, 26, 68, 38, 32, 68, 169, 22, 222, 38, 42, 50, 328, 40, 180, 219, 108, 26, 334, 82, 460, 82, 92, 72, 220, 108, 449, 86, 128, 80, 192, 22, 336, 110, 222, 218, 540, 84, 778, 129, 150, 80, 270, 54, 328, 356, 132, 68, 348, 22

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

			a(2) = |{9,21,63}| = 3, a(3) = |{7,14,28,52,56,91,104,182,364,728}| = 10, a(4) = |{13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095}| = 16,...

Cf. A059907-A059910, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{6, 3, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025*)

a(n) = if(n == 1, 0, numdiv(n^6-1) - numdiv(n^3-1) - numdiv(n^2-1) + numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^6-1)-tau(n^3-1)-tau(n^2-1)+tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A070920 a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.

Original entry on oeis.org

1, 15, 15, 65, 15, 225, 15, 175, 65, 225, 15, 975, 15, 225, 225, 369, 15, 975, 15, 975, 225, 225, 15, 2625, 65, 225, 175, 975, 15, 3375, 15, 671, 225, 225, 225, 4225, 15, 225, 225, 2625, 15, 3375, 15, 975, 975, 225, 15, 5535, 65, 975, 225, 975, 15, 2625, 225

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, 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.

Cf. A000005, A008683, A048691, A070919, A070921, A247516 (Mobius transform).

Join[{1},Table[Product[(k + 1)^4 - k^4, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

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

a(n) = vecprod(apply(x->(x+1)^4-x^4, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^4*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^4/k^s.
Multiplicative with a(p^e) = (e+1)^4 - e^4. - Amiram Eldar, Sep 03 2023

A070921 a(n) = Card{ (x,y,z,u,v) | lcm(x,y,z,u,v)=n }.

Original entry on oeis.org

1, 31, 31, 211, 31, 961, 31, 781, 211, 961, 31, 6541, 31, 961, 961, 2101, 31, 6541, 31, 6541, 961, 961, 31, 24211, 211, 961, 781, 6541, 31, 29791, 31, 4651, 961, 961, 961, 44521, 31, 961, 961, 24211, 31, 29791, 31, 6541, 6541, 961, 31, 65131, 211, 6541

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, 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.

Cf. A000005, A008683, A048691, A070919, A070920, A247517 (Mobius transform).

Join[{1},Table[Product[(k + 1)^5 - k^5, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^5*moebius(n/d)),","))

a(n) = vecprod(apply(x->(x+1)^5-x^5, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^5*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^5/k^s.
Multiplicative with a(p^e) = (e+1)^5 - e^5. - Amiram Eldar, Sep 03 2023

A086222 a(n) = card{ (x,y,z) | x <= y <= z and lcm(x,y,z) = n }.

Original entry on oeis.org

1, 3, 3, 6, 3, 13, 3, 10, 6, 13, 3, 30, 3, 13, 13, 15, 3, 30, 3, 30, 13, 13, 3, 54, 6, 13, 10, 30, 3, 71, 3, 21, 13, 13, 13, 73, 3, 13, 13, 54, 3, 71, 3, 30, 30, 13, 3, 85, 6, 30, 13, 30, 3, 54, 13, 54, 13, 13, 3, 178, 3, 13, 30, 28, 13, 71, 3, 30, 13, 71, 3, 135, 3, 13, 30, 30, 13, 71, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

A number of conjectures are possible, many of which should be easy to prove. Examples: (1) If n is a product of two primes then a(n)=13. (2) If n is a square of a prime then a(n)=6. - John W. Layman, Sep 01 2003

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048691, A070919, A018892, A086165, A377304, A008683.

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + 3 * Times @@ f2 @@@f + 2) / 6; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A048691(n) = numdiv(n^2);
A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
A086222(n) = ((A070919(n)+3*A048691(n)+2)/6); \\ Antti Karttunen, May 19 2017, after Jovovic's formula.

a(n) = {my(e = factor(n)[, 2]); (vecprod(apply(x->(x+1)^3-x^3, e)) + 3*vecprod(apply(x->2*x+1, e)) + 2) / 6;} \\ Amiram Eldar, Sep 03 2023

For a prime p, a(p) = 3.
a(n) = (A070919(n) + 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004
a(n) = Sum_{d|n} A377304(d)*A008683(n/d). - Ridouane Oudra, May 22 2025
a(n) = A086165(n) + A048691(n). - Ridouane Oudra, Aug 19 2025

More terms from John W. Layman, Sep 01 2003

A147810 Half the number of divisors of n^2+1.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 4, 1, 2, 1, 4, 3, 2, 1, 4, 2, 4, 1, 2, 1, 4, 2, 2, 2, 4, 3, 4, 2, 2, 1, 4, 3, 2, 1, 3, 2, 6, 2, 2, 2, 8, 2, 2, 2, 2, 2, 4, 1, 4, 1, 8, 2, 2, 2, 2, 2, 4, 2, 2, 1, 4, 4, 2, 3, 2, 4, 8, 1, 4, 2, 4, 2, 2, 2, 4, 3, 8, 1, 2, 2, 4, 2, 4, 1, 4, 2, 6, 1, 2, 2, 4, 4, 6

Offset: 1

M. F. Hasler, Dec 13 2008

For any n>0, n^2+1 cannot be a square and thus has an even number of divisors which always include 1 and n^2+1, therefore a(n) is always a positive integer.
Also number of ways to write n^2+1 as n^2+1 = x*y with 1 <= x <= y. - Michel Lagneau, Mar 10 2014
Also number of ways to write arctan(1/n) = arctan(1/x)+arctan(1/y), for integral 0 < n < x < y. - Matthijs Coster, Dec 09 2014
Number of ways that n can be expressed as (j*k-1)/(j+k) with j >= k > n. For any nonnegative integer n, the equation j*k = 1+n*(j+k) always has at least one integer solution with j >= k > n. As j >= k > n, let k=n+c (c is a positive integer), then j=n+(n^2+1)/c; we can easily conclude that c <= n, i.e., for n > 0, a(n) is the number of divisors of (n^2+1) which are <= n. - Zhining Yang, May 18 2023

			For n = 7 the a(7) = 3 solutions are (17,12), (32,9), (57,8). For  n = 13 the a(13) = 4 solutions are (30,23), (47,18), (98,15), (183,14). - _Zhining Yang_, May 18 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Shouen Wang, The general term formula of an integer sequence.

Cf. A048691, A093582, A290332, A290333, A359225.

with(numtheory); A147810:=n->tau(n^2+1)/2; seq(A147810(n), n=1..100); # Wesley Ivan Hurt, Mar 10 2014

Table[c=0; Do[If[i<=j && i*j==n^2+1, c++], {i, t=Divisors[n^2+1]}, {j, t}]; c, {n, 100}] (* Michel Lagneau, Mar 10 2014 *)

PARI
```
A147810(n)=numdiv(n^2+1)/2
```

from sympy import divisor_count
def A147810(n): return divisor_count(n**2+1)>>1 if n else 1 # Chai Wah Wu, Jul 09 2023

a(n) = A000005(A002522(n))/2 = A147809(n)+1.

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = 3/(2*Pi) = 0.477464... (A093582). - Amiram Eldar, Dec 01 2023

A348453 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1, 1, 158753814, 1, 7157114189

Offset: 1

N. J. A. Sloane, Oct 27 2021

The board has n^2 squares. The colors do not matter. The tiles are rook-connected polygons made from n^2/d_k squares.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
A348452 displays the same data in a less compact way. The present triangle is obtained by omitting the zero entries from A348452.
The data is taken from A004003, A172477, A348456, and Schutzman & MGGG (2018).
T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.

			The first eight rows of the triangle are:
  1,
  1, 2, 1,
  1, 10, 1,
  1, 70, 117, 36, 1,
  1, 4006, 1,
  1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1,
  1, 158753814, 1,
  1, 7157114189, ?, 187497290034, ?, ?, 1,
  ...
The corresponding divisors d_k are:
  1,
  1, 2, 4,
  1, 3, 9,
  1, 2, 4, 8, 16,
  1, 5, 25,
  ...
The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
N. J. A. Sloane, Illustration for T(3,2) = 10
N. J. A. Sloane, Illustration for T(4,2) = 70 [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
N. J. A. Sloane, Illustration for T(4,4) = 36 [Slide from an old talk of mine]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.

Cf. A348452. A348454 and A348455 are similar triangles with the data in each row reversed.
See also A004003, A099390, A172477, A348456.
Cf. A048691 (row lengths).

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

T(8,2) added May 04 2022 (see A348456) - N. J. A. Sloane, May 05 2022

cp's OEIS Frontend

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

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A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

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A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

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A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

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PARI

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A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

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PARI

Formula

A070920 a(n) = Card{ (x,y,z,u) | lcm(x,y,z,u)=n }.

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PARI

PARI

Formula

A070921 a(n) = Card{ (x,y,z,u,v) | lcm(x,y,z,u,v)=n }.

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PARI

PARI

Formula

A086222 a(n) = card{ (x,y,z) | x <= y <= z and lcm(x,y,z) = n }.

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