This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334080 #22 Apr 04 2024 10:12:14
%S A334080 1,2,2,3,2,4,2,4,3,4,2,6,3,4,4,5,2,6,2,6,4,4,2,8,3,6,4,6,2,8,2,6,4,5,
%T A334080 4,9,2,4,6,8,2,8,2,6,6,4,2,10,3,6,4,9,2,8,4,8,4,4,2,12,2,4,6,7,6,8,2,
%U A334080 8,4,9,2,12,2,4,6,6,4,12,2,10,5,4,2,12,4
%N A334080 Number of Pythagorean triples among the divisors of 60*n.
%C A334080 The odd numbers of the sequence are rare (see the table below).
%C A334080 The subsequence of odd terms begins with 1, 3, 3, 3, 5, 3, 5, 9, 3, 9, 7, 9, 5, 9, 9, 3, 11, 15, 5, 9, 5, 15, 9, 9, 9, 5, 19, 3, 15, 15, 9, ... (see the table at the link).
%C A334080 It is interesting to note that each set of divisors of A169823(n) contains m primitive Pythagorean triples for some n, m = 1, 2, ...
%C A334080 Examples:
%C A334080 - The set of divisors of A169823(1)= 60 contains only one primitive Pythagorean triple: (3, 4, 5).
%C A334080 - The set of divisors of A169823(136) = 8160 contains two primitive Pythagorean triples: (3, 4, 5) and (8, 15, 17).
%C A334080 - The set of divisors of A169823(910) = 54600 contains three primitive Pythagorean triples: (3, 4, 5), (5, 12, 13) and (7, 24, 25).
%C A334080 There is an interesting property: we observe that a(n) = A000005(n) except for n in the set {13, 26, 34, 39, 52, 65, 68, 70, 78, 91, 102, ...}. This set contains subset of numbers of the form 13*k, 34*k, 70*k, 203*k, 246*k, 259*k, ... for k = 1, 2, ...
%C A334080 We recognize the sequence A081752: {13, 34, 70, 203, 246, 259, 671, ...} (ordered product of the sides of primitive Pythagorean triangles divided by 60).
%C A334080 The following table shows the numbers of odd terms < 10^k for k = 2, 3, 4, 5, 6 and 7. For instance, among the 16 multiples of 60 less than 10^3, the divisors of the five numbers 60, 240, 540, 780 and 960 contain 1, 3, 3, 3 and 5 Pythagorean triples respectively, and that represents 31.25% of odd numbers.
%C A334080 +---------------+-----------------+---------------------+----------+
%C A334080 | Intervals | Number of | Number of odd terms | |
%C A334080 | D(k) < 10^k | multiples of 60 | in D(k) | % |
%C A334080 | k = 2,3,...,7 | in D(k) | | |
%C A334080 +---------------+-----------------+---------------------+----------+
%C A334080 | < 10^2 | 1 | 1 | 100% |
%C A334080 | < 10^3 | 16 | 5 | 31.250% |
%C A334080 | < 10^4 | 166 | 18 | 10.843% |
%C A334080 | < 10^5 | 1666 | 72 | 4.321% |
%C A334080 | < 10^6 | 16666 | 256 | 1.536% |
%C A334080 | < 10^7 | 166666 | 879 | 0.527% |
%C A334080 |---------------+-----------------+---------------------+----------+
%H A334080 Michel Lagneau, <a href="/A334080/a334080.pdf">Odd terms</a>.
%e A334080 a(4) = 3 because the divisors of A169823(4) = 240 are {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} with 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.
%p A334080 with(numtheory):
%p A334080 for n from 60 by 60 to 5400 do :
%p A334080 d:=divisors(n):n0:=nops(d):it:=0:
%p A334080 for i from 1 to n0-1 do:
%p A334080 for j from i+1 to n0-2 do :
%p A334080 for m from i+2 to n0 do:
%p A334080 if d[i]^2 + d[j]^2 = d[m]^2
%p A334080 then
%p A334080 it:=it+1:
%p A334080 else
%p A334080 fi:
%p A334080 od:
%p A334080 od:
%p A334080 od:
%p A334080 printf(`%d, `,it):
%p A334080 od:
%o A334080 (PARI) ishypo(n) = setsearch(Set(factor(n)[, 1]%4), 1); \\ A009003
%o A334080 a(n) = {n *= 60; my(d=divisors(n), nb=0); for (i=3, #d, if (ishypo(d[i]), for (j=2, i-1, for (k=3, j-1, if (d[j]^2 + d[k]^2 == d[i]^2, nb++););););); nb;} \\ _Michel Marcus_, Apr 26 2020
%Y A334080 Cf. A000005, A009003, A014442, A081752, A089120, A144255, A169823.
%K A334080 nonn
%O A334080 1,2
%A A334080 _Michel Lagneau_, Apr 14 2020