A335654 Numbers m such that the elements of all Pythagorean quadruples belonging to the set of divisors are exactly their first k divisors for some k.

504, 1008, 1512, 1872, 2016, 3024, 3528, 3744, 4032, 4536, 5616, 6048, 6552, 7056, 7488, 8064, 9072, 9576, 10584, 11232, 12096, 13104, 13608, 14112, 14976, 16128, 16848, 17784, 18144, 19152, 19656, 21168, 21672, 22464, 23688, 24192, 24336, 24696, 26208, 27216

Offset: 1

Author

Michel Lagneau, Jun 16 2020

Keywords

• nonn

Comments

Members m in A330893 for which there exists a number k < tau(m) such that the elements of all Pythagorean quadruples included in the set of the divisors of m are the first k divisors of m.

Conjecture 1: a(n) == 0 (mod 72).

Conjecture 2: if the numbers m such that the elements of all Pythagorean quadruples contained in the set of divisors of m are exactly the first k divisors of m, then k = tau(m) - 4 or tau(m) - 5.

The corresponding k of the sequence are given by the sequence {b(n)} = {20, 26, 28, 25, 32, 36, 32, 31, 38, 36, 35, 44, 44, 41,...} and the sequence {c(n)} = {tau(a(n)) - b(n)} = {4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4,...}. We observe that c(n) = 4 or 5 (see the table in the link). For n = 1, 2,...,400, the statistic observed is 301 occurrences for the number 4 (75.25 %) and 99 occurrences for the number 5 (24.75 %). It is probable that Pr(4) tends to .75 and Pr(5) tends to .25 when n tends into infinity, where Pr(x) is the probability of the occurrence x.

Assumes the elements in the quadruple are distinct. Otherwise 6, 12, 18, 24, ... are also terms. For instance the divisors of 6 are 1,2,3,6 and 1^2 + 2^2 + 2^2 = 3^2. - Chai Wah Wu, Nov 16 2020

Examples

			504 is in the sequence because the divisors are  {1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504} and the elements of the 8 Pythagorean quadruples belonging to the set of divisors of 504: (1, 4, 8, 9), (2, 3, 6, 7), (4, 6, 12, 14), (6, 9, 18, 21), (7, 28, 56, 63), (8, 12, 24, 28), (12, 18, 36, 42) and (24, 36, 72, 84) are the first 20 divisors of 504 with 20 = tau(504) - 4 = 24 - 4.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000
• Michel Lagneau, Table

Crossrefs

Cf. A000005, A330893, A330894, A331365.

Programs

Maple

with(numtheory):
for n from 6 by 6 to 20000 do :lst:={}:lst1:={}:
   d:=divisors(n):n0:=nops(d):
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^2 + d[j]^2 + d[k]^2 = d[m]^2
        then
        lst:=lst union {d[i]} union {d[j]} union {d[k]} union {d[m]}:
        else
       fi:
      od:
     od:
    od:
    od:
    n1:=nops(lst):
     for l from 1 to n1 do:
      lst1:= lst1 union {d[l]}:
     od:
    if lst=lst1 and lst<>{}
     then
     printf(`%d, `,n):
    else fi:
   od: