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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.

%I A335654 #21 Aug 31 2021 02:43:58
%S A335654 504,1008,1512,1872,2016,3024,3528,3744,4032,4536,5616,6048,6552,7056,
%T A335654 7488,8064,9072,9576,10584,11232,12096,13104,13608,14112,14976,16128,
%U A335654 16848,17784,18144,19152,19656,21168,21672,22464,23688,24192,24336,24696,26208,27216
%N 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.
%C A335654 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.
%C A335654 Conjecture 1: a(n) == 0 (mod 72).
%C A335654 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.
%C A335654 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.
%C A335654 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
%H A335654 Chai Wah Wu, <a href="/A335654/b335654.txt">Table of n, a(n) for n = 1..1000</a>
%H A335654 Michel Lagneau, <a href="/A335654/a335654_1.pdf">Table</a>
%e A335654 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.
%p A335654 with(numtheory):
%p A335654 for n from 6 by 6 to 20000 do :lst:={}:lst1:={}:
%p A335654    d:=divisors(n):n0:=nops(d):
%p A335654     for i from 1 to n0-3 do:
%p A335654      for j from i+1 to n0-2 do :
%p A335654       for k from j+1 to n0-1 do:
%p A335654       for m from k+1 to n0 do:
%p A335654        if d[i]^2 + d[j]^2 + d[k]^2 = d[m]^2
%p A335654         then
%p A335654         lst:=lst union {d[i]} union {d[j]} union {d[k]} union {d[m]}:
%p A335654         else
%p A335654        fi:
%p A335654       od:
%p A335654      od:
%p A335654     od:
%p A335654     od:
%p A335654     n1:=nops(lst):
%p A335654      for l from 1 to n1 do:
%p A335654       lst1:= lst1 union {d[l]}:
%p A335654      od:
%p A335654     if lst=lst1 and lst<>{}
%p A335654      then
%p A335654      printf(`%d, `,n):
%p A335654     else fi:
%p A335654    od:
%Y A335654 Cf. A000005, A330893, A330894, A331365.
%K A335654 nonn
%O A335654 1,1
%A A335654 _Michel Lagneau_, Jun 16 2020