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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.

%I A336432 #46 Apr 21 2024 19:14:49
%S A336432 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,
%T A336432 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,
%U A336432 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
%N 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.
%C A336432 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}.
%C A336432 Every term of the sequence is repeated indefinitely; for instance:
%C A336432 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).
%C A336432 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).
%C A336432 a(n) = 5 for n = 32, 243, 3125, 16807, ... (fifth powers of primes; see A050997).
%C A336432 a(n) = 15 for n = 64, 729, 15625, 117649, ... (numbers with 7 divisors: 6th powers of primes; see A030516).
%H A336432 Antti Karttunen, <a href="/A336432/b336432.txt">Table of n, a(n) for n = 1..10000</a>
%H A336432 Antti Karttunen, <a href="/A336432/a336432.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A336432 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e A336432 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).
%p A336432 with(numtheory):nn:=100:
%p A336432 for n from 1 to nn do:
%p A336432 it:=0:d:=divisors(n):n0:=nops(d):
%p A336432   for i from 1 to n0-3 do:
%p A336432    for j from i+1 to n0-2 do:
%p A336432      for k from j+1 to n0-1 do:
%p A336432        for l from k+1 to n0 do:
%p A336432     if igcd(d[i],d[j],d[k],d[l])> 1
%p A336432        then
%p A336432        it:=it+1:
%p A336432        else
%p A336432       fi:
%p A336432      od:
%p A336432     od:
%p A336432    od:
%p A336432   od:
%p A336432     printf(`%d, `,it):
%p A336432 od:
%t A336432 Array[Count[GCD @@ # & /@ Subsets[Divisors[#], {4}], _?(# > 1 &)] &, 100] (* _Amiram Eldar_, Oct 31 2020 after _Michael De Vlieger_ at A336530 *)
%o A336432 (PARI) 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
%o A336432 (PARI) 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
%Y A336432 Cf. A275387, A336530.
%K A336432 nonn
%O A336432 1,24
%A A336432 _Michel Lagneau_, Oct 05 2020
%E A336432 Terms corrected by _David A. Corneth_, Oct 31 2020