A336530 -id:A336530 - cp's OEIS frontend

A336432 Number of ordered quadruples of divisors (d_i, d_j, d_k, d_m) of n such that GCD(d_i, d_j, d_k, d_m) > 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 16, 0, 0, 0, 1, 0, 3, 0, 5, 0, 0, 0, 29, 0, 0, 0, 16, 0, 3, 0, 1, 1, 0, 0, 74, 0, 1, 0, 1, 0, 16, 0, 16, 0, 0, 0, 98, 0, 0, 1, 15, 0, 3, 0, 1, 0, 3, 0, 181, 0, 0, 1, 1, 0, 3, 0, 74, 1, 0, 0, 98, 0, 0, 0, 16, 0, 98, 0, 1, 0, 0, 0, 220, 0, 1, 1, 29, 0, 3, 0

Offset: 1

Michel Lagneau, Oct 05 2020

nonn

Number of elements in the set {(x, y, z, w): x|n, y|n, z|n, w|n, x < y < z < w, GCD(x, y, z, w) > 1}.
Every term of the sequence is repeated indefinitely; for instance:
a(n) = 0 for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, ... (numbers k such that product of proper divisors of k is <= k; i.e., product of divisors of k is <= k^2; see A007964).
a(n) = 1 for n = 12, 16, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 81, 92, 98, 99, ... (either 4th power of a prime, or product of a prime and the square of a different prime; see A080258).
a(n) = 5 for n = 32, 243, 3125, 16807, ... (fifth powers of primes; see A050997).
a(n) = 15 for n = 64, 729, 15625, 117649, ... (numbers with 7 divisors: 6th powers of primes; see A030516).

			a(30) = 3 because the divisors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} and GCD(d_i, d_j, d_k, d_m) > 1 for the following 3 quadruples of divisors: (2,6,10,30), (3,6,15,30) and (5,10,15,30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A275387, A336530.

with(numtheory):nn:=100:
for n from 1 to nn do:
it:=0:d:=divisors(n):n0:=nops(d):
  for i from 1 to n0-3 do:
   for j from i+1 to n0-2 do:
     for k from j+1 to n0-1 do:
       for l from k+1 to n0 do:
    if igcd(d[i],d[j],d[k],d[l])> 1
       then
       it:=it+1:
       else
      fi:
     od:
    od:
   od:
  od:
    printf(`%d, `,it):
od:

Array[Count[GCD @@ # & /@ Subsets[Divisors[#], {4}], ?(# > 1 &)] &, 100] (* _Amiram Eldar, Oct 31 2020 after Michael De Vlieger at A336530 *)

a(n) = my(d=divisors(n)); sum(i=1, #d-3, sum (j=i+1, #d-2, sum (k=j+1, #d-1, sum (m=k+1, #d, gcd([d[i], d[j], d[k], d[m]]) > 1)))); \\ Michel Marcus, Oct 31 2020

a(n) = {my(f = factor(n), vp = vecprod(f[,1]), d = divisors(vp), res = 0); for(i = 2, #d, res-=binomial(numdiv(n/d[i]), 4)*(-1)^omega(d[i])); res} \\ David A. Corneth, Oct 31 2020

Terms corrected by David A. Corneth, Oct 31 2020

A338509 a(n) is the number of ordered triples of divisors d_i < d_j < d_k of m such that GCD(d_i, d_j, d_k) > 1 where m is the least number having its prime signature; m = A025487(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 4, 23, 12, 10, 36, 62, 87, 20, 120, 130, 289, 35, 284, 432, 235, 200, 356, 682, 56, 555, 1256, 385, 1005, 795, 1330, 84, 960, 2775, 588, 2939, 1501, 1844, 2297, 120, 3436, 1526, 4304, 1720, 5205, 852, 6514, 2538, 5001, 3647, 165, 7341, 2280, 2280, 11712

Offset: 1

David A. Corneth, Oct 31 2020

Primitive sequence to A336530 as that sequence only depends on the prime signature of n.

			a(6) = 12 as A025487(6) = 12 and there are 5 triples of divisors of 12 (x, y, z) such that g = gcd(x, y, z) are 12. 4 of them have g = 2 as 12/2 = 6 has 4 divisors and binomial(4, 3) = 4, 1 of them has g = 3 as 12/3 = 4 has 3 divisors and binomial(3, 3) = 1 and 0 of them have g = 6 as 12/6 = 2 has 3 divisors and binomial(2, 3) = 0.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A025487, A336530.

a(n) = A336530(A025487(n)).

cp's OEIS Frontend

A336432 Number of ordered quadruples of divisors (d_i, d_j, d_k, d_m) of n such that GCD(d_i, d_j, d_k, d_m) > 1.

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