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.

28, 84, 120, 240, 360, 252, 210, 660, 1008, 1848, 630, 1320, 420, 2310, 840, 4830, 1680, 3360, 5880, 11700, 1980, 4200, 1260, 9660, 3960, 3780, 2520, 6930, 4620, 8190, 6300, 7560, 5040, 18900, 19320, 5460, 23760, 7140, 39600, 15120, 27300, 12600, 59220, 45360

Offset: 1

Author

Michel Lagneau, Feb 14 2019

Keywords

• nonn

Comments

We observe that a(n) == 0 (mod 6) for n > 1, and a(n) == 0 (mod 30) for n > 10.

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)}.

The following table gives the first 10 subsequences E(q).

+----+--------------------------------------------+

| q | E(q) such that q*E(q) is a subsequence |

+----+--------------------------------------------+

| 2 | {120, 210, 420, 630, 660, 840, 1260, ...} |

| 3 | {28, 84, 120, 210, 420, 660, 840, ...} |

| 4 | {210, 252, 420, 630, 840, 1260, 3780, ...} |

| 5 | {84, 252, 840, 1008, 1260, 2520, ...} |

| 6 | {210, 420, 630, 660, 840, 1260, 2520, ...} |

| 7 | {120, 240, 360, 660, 840, ...} |

| 8 | {210, 420, 630, ...} |

| 9 | {28, 420, 840, 1680, 5040, ...} |

| 10 | {84, 252, 420, 630, 1260, 3960, ...} |

+----+--------------------------------------------+

Examples

			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).
		

Crossrefs

Cf. A000005, A027750, A080257.

Programs

Maple

with(numtheory):nn:=1000:
for n from 1 to nn do:
ii:=0:it:=0:
for k from 1 to 10^5 while(ii=0) do:
d:=divisors(k):n0:=nops(d):it:=0:
for a from 1 to n0-1 do:
  for b from a+1 to n0 do:
   lst1:={d[a]} union {d[b]}:lst:= d minus lst1:n1:=nops(lst):
     for i from 1 to n1-1 do:
       for j from i+1 to n1 do:
         if d[a]*d[b]-lst[i]*lst[j]=1
         then
          it:=it+1:
          else fi:
        od:
       od:
       od:
      od:
      if it=n then ii:=1:printf (`%d %d \n`,n,k):
      else fi:
      od:
     od: