A379647 Let D(k) = {d(k,i)}, i = 1,2,...,q be the set of q divisors of an integer k. a(n) is the smallest number k such that there exist exactly n distinct integers M, 1 < M <= k, where each set D(k) mod M = {0,1,2,...,M-1}.

1, 2, 6, 12, 84, 60, 120, 420, 2280, 840, 3360, 2520, 5040, 10080, 13860, 21840, 32760, 55440, 65520, 98280, 163800, 196560, 491400, 556920, 360360, 720720, 1670760, 1801800, 1081080, 2162160, 4324320, 5266800, 6126120, 12612600, 10450440, 12252240, 17907120

Offset

0,2

Comments

a(0) = 1 by convention. We observe that a(n) == 0 (mod 6) for n >= 2.

Links

Table of n, a(n) for n=0..36.

Example

+----+----+---------------------------------+-----------------------------+
|  n |  k |         divisors of k           |    D(k) mod M               |
+----+----+---------------------------------+-----------------------------+
|  0 |  1 | {1}                             |  x x x x x x x              |
+----+----+---------------------------------+-----------------------------+
|  1 |  2 | {1,2}                           | {0,1} mod 2                 |
+----+----+---------------------------------+-----------------------------+
|  2 |  6 | {1,2,3,6}                       | {0,1} mod 2, {0,1,2} mod 3  |
+----+----+---------------------------------+-----------------------------+
|  3 | 12 | {1,2,3,4,6,12}                  | {0,1} mod 2, {0,1,2} mod 3, |
|    |    |                                 | {0,1,2,3} mod 4             |
+----+----+---------------------------------+-----------------------------+
|  4 | 84 | {1,2,3,4,6,7,12,14,21,28,42,84} | {0,1} mod 2, {0,1,2} mod 3, |
|    |    |                                   {0,1,2,3} mod 4,            |
|    |    |                                 | {0,1,2,3,4,5,6} mod 7       |
+----+----+---------------------------------+-----------------------------+
|  5 | 60 | {1,2,3,4,5,6,10,12,15,20,30,60} | {0,1} mod 2, {0,1,2} mod 3, |
|    |    |                                 | {0,1,2,3} mod 4,            |
|    |    |                                 | {0,1,2,3,4} mod 5,          |
|    |    |                                 | {0,1,2,3,4,5} mod 6         |
+----+----+---------------------------------+-----------------------------+

Maple

with(numtheory):
nn:=10^7:
for n from 0 to nn do:
 ii:=0:
 for k from 1 to nn while(ii=0) do:
  x:=divisors(k):n0:=nops(x):ind:=0:
    for M from 2 to k do:
      lst:={}:
        for i from 1 to n0 do:
         lst:=lst union {irem(x[i], M)}:
        od:
         lst1:={}:jj:=0:n1:=nops(lst):
           for j from 0 to n1-1 do:
            if lst[j+1]=j then jj:=jj+1: elst fi:
            od:
             if jj=M
             then ind:=ind+1:else fi:
       od:
    if ind=n then ii:=1:printf(`%d %d \n`, n, k):
    else fi:
 od:
od:

Crossrefs

Cf. A027750, A379645.

Sequence in context: A144144 A226178 A129085 * A274941 A141288 A038787

Adjacent sequences: A379644 A379645 A379646 * A379648 A379649 A379650

Keyword

nonn

Author

Michel Lagneau, Dec 28 2024

Extensions

More terms from Jinyuan Wang, Jan 11 2025

Status

approved