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, ...} |
+----+--------------------------------------------+
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 14 2019
STATUS
approved