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

1,1

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, ...} |

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

Links

Table of n, a(n) for n=1..44.

Example

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).

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:

Crossrefs

Cf. A000005, A027750, A080257.

Sequence in context: A126382 A165009 A179790 * A254145 A271735 A280885

Adjacent sequences: A306424 A306425 A306426 * A306428 A306429 A306430

Keyword

nonn

Author

Michel Lagneau, Feb 14 2019

Status

approved