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A306427 Least integer m such that there are exactly n quadruples of distinct divisors (d_i, d_j, d_k, d_l) among the divisors of m having the property d_i * d_j - d_k * d_l = 1, for some i, j, k, l.

%I A306427 #12 Oct 13 2019 18:05:18
%S A306427 28,84,120,240,360,252,210,660,1008,1848,630,1320,420,2310,840,4830,
%T A306427 1680,3360,5880,11700,1980,4200,1260,9660,3960,3780,2520,6930,4620,
%U A306427 8190,6300,7560,5040,18900,19320,5460,23760,7140,39600,15120,27300,12600,59220,45360
%N A306427 Least integer m such that there are exactly n quadruples of distinct divisors (d_i, d_j, d_k, d_l) among the divisors of m having the property d_i * d_j - d_k * d_l = 1, for some i, j, k, l.
%C A306427 We observe that a(n) == 0 (mod 6) for n > 1, and a(n) == 0 (mod 30) for n > 10.
%C A306427 Conjecture: for each integer q > 1, there exists a subsequence E(q) of {a(n)} such that q*E(q) is also a subsequence of {a(n)}.
%C A306427 The following table gives the first 10 subsequences E(q).
%C A306427 +----+--------------------------------------------+
%C A306427 |  q |   E(q) such that q*E(q) is a subsequence   |
%C A306427 +----+--------------------------------------------+
%C A306427 |  2 | {120, 210, 420, 630, 660, 840, 1260, ...}  |
%C A306427 |  3 | {28, 84, 120, 210, 420, 660, 840, ...}     |
%C A306427 |  4 | {210, 252, 420, 630, 840, 1260, 3780, ...} |
%C A306427 |  5 | {84, 252, 840, 1008, 1260, 2520, ...}      |
%C A306427 |  6 | {210, 420, 630, 660, 840, 1260, 2520, ...} |
%C A306427 |  7 | {120, 240, 360, 660, 840, ...}             |
%C A306427 |  8 | {210, 420, 630, ...}                       |
%C A306427 |  9 | {28, 420, 840, 1680, 5040, ...}            |
%C A306427 | 10 | {84, 252, 420, 630, 1260, 3960, ...}       |
%C A306427 +----+--------------------------------------------+
%e A306427 a(7) = 210 because the divisors of 210 are {1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210} with seven following quadruples (1, 7, 2, 3), (1, 15, 2, 7), (1, 21, 2, 10), (2, 3, 1, 5), (3, 5, 1, 14), (3, 5, 2, 7) and (3, 7, 2, 10).
%p A306427 with(numtheory):nn:=1000:
%p A306427 for n from 1 to nn do:
%p A306427 ii:=0:it:=0:
%p A306427 for k from 1 to 10^5 while(ii=0) do:
%p A306427 d:=divisors(k):n0:=nops(d):it:=0:
%p A306427 for a from 1 to n0-1 do:
%p A306427   for b from a+1 to n0 do:
%p A306427    lst1:={d[a]} union {d[b]}:lst:= d minus lst1:n1:=nops(lst):
%p A306427      for i from 1 to n1-1 do:
%p A306427        for j from i+1 to n1 do:
%p A306427          if d[a]*d[b]-lst[i]*lst[j]=1
%p A306427          then
%p A306427           it:=it+1:
%p A306427           else fi:
%p A306427         od:
%p A306427        od:
%p A306427        od:
%p A306427       od:
%p A306427       if it=n then ii:=1:printf (`%d %d \n`,n,k):
%p A306427       else fi:
%p A306427       od:
%p A306427      od:
%Y A306427 Cf. A000005, A027750, A080257.
%K A306427 nonn
%O A306427 1,1
%A A306427 _Michel Lagneau_, Feb 14 2019