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A377418 a(n) is the smallest integer k whose set of divisors contains exactly n triples (x,y,z) of distinct divisors considered as integer-sided triangles with integer areas, or 0 if no such k exists.

%I A377418 #4 Nov 18 2024 07:38:00
%S A377418 60,120,240,360,960,720,3480,1440,1680,2880,6600,2520,4200,10440,5460,
%T A377418 6240,4680,5040,20400,7800,18360,17160,26520,10080,47040,9360,15120,
%U A377418 10920,55080,20160,15600,16380,34320,33600,18720,27300,165240,53040,37800,25200,21840
%N A377418 a(n) is the smallest integer k whose set of divisors contains exactly n triples (x,y,z) of distinct divisors considered as integer-sided triangles with integer areas, or 0 if no such k exists.
%C A377418 We observe that this sequence is a subsequence of A169823: a(n) == 0 (mod 60).
%C A377418 The area A of a triangle whose sides have lengths x, y, and z is given by Heron's formula: A = sqrt(s*(s-x)*(s-y)*(s-z)), where s = (x+y+z)/2.
%e A377418 a(3) = 240 because there are 3 triples of divisors (3, 4, 5), (6, 8, 10) and (12, 16, 20) with integer areas 36, 576, 9216 respectively (Pythagorean triples). The first triple is primitive.
%e A377418 a(9)=1680 because there are 9 triples of divisors (3,4,5), (6,8,10), (7,15,20), (12,16,20), (14,30,40), (21,28,35), (28,60,80), (42,56,70), (84,112,140) with 5 Pythagorean triples : (3,4,5), (6,8,10), (21,28,35), (42,56,70), (84,112,70). The other 4 triangles are arbitrary.
%p A377418 with(numtheory):
%p A377418 for n from 1 to 41 do:
%p A377418 ii:=0:
%p A377418 for m from 4 to 10^7 while(ii=0) do:it:=0:
%p A377418  d:=divisors(m):n0:=nops(d):
%p A377418   for i from 2 to n0-2 do:
%p A377418    for j from i+1 to n0 do:
%p A377418      for k from j+1 to n0 do:
%p A377418        x:=d[i]:y:=d[j]:z:=d[k]:s:=(x+y+z)/2:A:=s*(s-x)*(s-y)*(s-z):
%p A377418        if A>0 and sqrt(A)=floor(sqrt(A)) then it:=it+1:else
%p A377418         fi:
%p A377418        od:
%p A377418     od:
%p A377418    od:
%p A377418     if it=n then printf(`%d %d \n`,it,m):ii:=1:
%p A377418      else fi:
%p A377418   od:
%p A377418 od:
%Y A377418 Cf. A169823, A188158, A334382.
%K A377418 nonn
%O A377418 1,1
%A A377418 _Michel Lagneau_, Oct 27 2024