A306427 - OEIS

%I #12 Oct 13 2019 18:05:18

%S 28,84,120,240,360,252,210,660,1008,1848,630,1320,420,2310,840,4830,

%T 1680,3360,5880,11700,1980,4200,1260,9660,3960,3780,2520,6930,4620,

%U 8190,6300,7560,5040,18900,19320,5460,23760,7140,39600,15120,27300,12600,59220,45360

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

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

%C +----+--------------------------------------------+

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

%C +----+--------------------------------------------+

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

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

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

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

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

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

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

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

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

%C +----+--------------------------------------------+

%e 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 with(numtheory):nn:=1000:

%p for n from 1 to nn do:

%p ii:=0:it:=0:

%p for k from 1 to 10^5 while(ii=0) do:

%p d:=divisors(k):n0:=nops(d):it:=0:

%p for a from 1 to n0-1 do:

%p   for b from a+1 to n0 do:

%p    lst1:={d[a]} union {d[b]}:lst:= d minus lst1:n1:=nops(lst):

%p      for i from 1 to n1-1 do:

%p        for j from i+1 to n1 do:

%p          if d[a]*d[b]-lst[i]*lst[j]=1

%p          then

%p           it:=it+1:

%p           else fi:

%p         od:

%p        od:

%p        od:

%p       od:

%p       if it=n then ii:=1:printf (`%d %d \n`,n,k):

%p       else fi:

%p       od:

%p      od:

%Y Cf. A000005, A027750, A080257.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 14 2019