A089233 -id:A089233 - cp's OEIS frontend

A343660 Number of maximal pairwise coprime sets of at least two 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, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

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

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343652(n) - A005361(n).

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 87, 138, 211, 324, 486, 727, 1079, 1584, 2305, 3337, 4789, 6830, 9712, 13689, 19225, 26841, 37322, 51598, 71108, 97580, 133350, 181558, 246335, 332991, 448706, 602607, 806732, 1077333, 1433885, 1903682, 2520246, 3328549, 4383929

Offset: 0

Gus Wiseman, May 25 2018

nonn

			The a(6) = 21 unitary factorizations:
(13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
(2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
(2*3*5)
The a(6) = 21 multiset partitions:
{{6}}
{{2,4}}
{{1,5}}
{{3,3}}
{{2,2,2}}
{{1,1,4}}
{{1,2,3}}
{{1,1,2,2}}
{{1,1,1,3}}
{{1,1,1,1,2}}
{{1,1,1,1,1,1}}
{{1},{5}}
{{1},{2,3}}
{{2},{4}}
{{2},{1,3}}
{{2},{1,1,1,1}}
{{1,1},{4}}
{{1,1},{2,2}}
{{3},{1,2}}
{{3},{1,1,1}}
{{1},{2},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

Row sums of A321878.

Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz in A321878 *)

A284167 a(n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.

Original entry on oeis.org

2, 5, 7, 10, 8, 15, 8, 18, 16, 18, 10, 29, 8, 19, 25, 28, 10, 33, 10, 35, 26, 20, 12, 50, 18, 20, 31, 36, 12, 51, 10, 42, 27, 23, 33, 62, 8, 22, 30, 60, 12, 53, 10, 40, 52, 22, 14, 78, 20, 41, 28, 38, 12, 63, 36, 63, 30, 24, 16, 95, 8, 23, 59, 60, 32, 54, 10

Offset: 1

Ctibor O. Zizka, Mar 21 2017

nonn

Let S(n,n) be the number of solutions of the equation n/x + n/y = c where n, c, x, and y are positive integers. Then S(n,n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.
For c = 1 , S(n,n) = A000005(n).
Let S(n,m) be the number of solutions of the equation n/x + m/y = c where n, m, c, x, and y are positive integers, n not equal to m. Let k(i) be the i-th divisor of n, and k(j) the j-th divisor of m. Let d(t) be the number of divisors of t. Let R = d(k(i) + k(j)). Then S(n,m) = Sum_{i=1..A000005(n)} Sum_{j=1..A000005(m)} [R*1 if gcd(k(i),k(j)) = 1 , R*0 else].
For c = 1 , S(n,m) = A000005(n) * A000005(m) - P, where P is the number of divisor pairs such that gcd(k(i),k(j)) >= 2.

			For n = 4, divisors of 4 are 1, 2, 4; thus a(4) = d(4+1) + d(4+2) + d(4+4) = d(5) + d(6) + d(8) = 2 + 4 + 4 = 10.

R. L. Graham, Paul Erdos and Egyptian Fractions, Bolyai Society Mathematical Studies 25, pp 289-309, 2013.

Cf. A000005, A018892, A063647, A089233, A275387.

a[n_] := Sum[DivisorSigma[0, d + n], {d, Divisors@n}]; Array[a, 67] (* Giovanni Resta, Mar 21 2017 *)

for(n=1, 101, print1(sumdiv(n, d, numdiv(d + n)),", ")) \\ Indranil Ghosh, Mar 22 2017

from sympy import divisor_count, divisors
def a(n):
    return sum(divisor_count(n + d) for d in divisors(n)) # Indranil Ghosh, Mar 22 2017

a(21)-a(67) from Giovanni Resta, Mar 21 2017

A321519 Let d(n,i), i = 1..k be the k divisors of n^2 + 1 (the number 1 is not counted). a(n) is the number of ordered pairs d(n,i) < d(n,j) such that gcd(d(n,i), d(n,j)) = 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 6, 0, 1, 0, 6, 2, 1, 0, 6, 1, 6, 0, 1, 0, 6, 1, 1, 1, 6, 2, 6, 1, 1, 0, 6, 2, 1, 0, 2, 1, 11, 1, 1, 1, 25, 1, 1, 1, 1, 1, 6, 0, 6, 0, 16, 1, 1, 1, 1, 1, 6, 1, 1, 0, 6, 3, 1, 2, 1, 6, 25, 0, 6, 1, 6, 1, 1, 1, 6, 2, 25, 0, 1, 1

Offset: 1

Michel Lagneau, Nov 12 2018

nonn

Terms only depends on prime signature of n^2+1. - David A. Corneth, Nov 14 2018
We observe an interesting statistic for n <= 10^5: the four values of a(n) = 0, 1, 6, 25 represent more than 82% (see the table below).
a(A005574(n)) = 0, a(A085722(n)) = 1, a(A272078(n)) = 6, a(A316351(n)) = 25.
In the general case, a(k) = m if k^2+1 = p*q^m, m = 1, 2, 3, ... with p, q primes.
+--------------+-----------------------+------------+
|              | number of occurrences |            |
|     a(n)     |     for n <= 10^5     | percentage |
+--------------+-----------------------+------------+
|       0      |          6656         |    6.656%  |
|       1      |         23255         |   23.255%  |
|       6      |         31947         |   31.947%  |
|      25      |         20461         |   20.461%  |
| other values |         17681         |   17.681%  |
+--------------+-----------------------+------------+

			a(13) = 6 because the divisors {d(i)} of 13^2 + 1 = 170 (without the number 1)  are  {2, 5, 10, 17, 34, 85, 170}, and gcd(d(i), d(j)) = 1 for the 6 following pairs of elements of {d(i)}: (2, 5), (2, 17), (2, 85), (5, 17), (5, 34) and (10, 17).

Cf. A005574, A002522, A085722, A089233, A193432, A272078, A316351.

with(numtheory):nn:=10^3:
for n from 1 to nn do:
  it:=0:d:=divisors(n^2+1):n0:=nops(d):
   for k from 2 to n0-1 do:
    for l from k+1 to n0 do:
     if gcd(d[k],d[l])= 1
      then
      it:=it+1
      else
     fi:
   od:
  od:
  printf(`%d, `,it):
od:

f[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Map[f, Range[0,100]^2+1] (* Amiram Eldar, Nov 14 2018 after Robert G. Wilson v at A089233 *)

a(n) = {my(d=divisors(n^2+1)); sum(k=2, #d, sum(j=2, k-1, gcd(d[k], d[j]) == 1));} \\ Michel Marcus, Nov 12 2018

a(n) = A089233(n^2+1). - Michel Marcus, Nov 13 2018

cp's OEIS Frontend

A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

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A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

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A284167 a(n) = Sum_{i=1..A000005(n)} d(n+k(i)), where d(t) is the number of divisors of t and k(i) is the i-th divisor of n.

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A321519 Let d(n,i), i = 1..k be the k divisors of n^2 + 1 (the number 1 is not counted). a(n) is the number of ordered pairs d(n,i) < d(n,j) such that gcd(d(n,i), d(n,j)) = 1.

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