cp's OEIS Frontend

A297294 Number of primitive Pythagorean triples (PPTs) that have 2^n-1 as the length of their odd leg where n is the sequence index.

%I A297294 #45 Jan 27 2018 13:44:19
%S A297294 0,1,1,2,1,2,1,4,2,4,2,8,1,4,4,8,1,8,1,16,4,8,2,32,4,4,4,32,4,32,1,16,
%T A297294 8,4,8,128,2,4,8,64,2,32,4,64,32,8,4,256,2,64,16,64,4,32,32,128,8,32,
%U A297294 2,1024,1,4,32,64,4,128,2,64,8,256,4,2048,4,16,64,64,8,64,4,256,32,16,2,2048,4,16,32,512,1,1024
%N A297294 Number of primitive Pythagorean triples (PPTs) that have 2^n-1 as the length of their odd leg where n is the sequence index.
%C A297294 2^n-1 for n = 0 and 1 give the Mersenne numbers 0 and 1, neither of which can be the side length of a PPT. For n > 1, all Mersenne numbers are congruent to 3 mod 4. Consequently, no Mersenne number can be the length of the hypotenuse of a PPT.
%C A297294 If 2^n-1 is the length of the odd leg of a PPT its divisors can provide a set of pairs {x, y} such that for each pair, x*y = 2^n-1, x < y and gcd(x, y) = 1. Using Euclid's parametric generators for PPTs (s^2+t^2, 2s*t, s^2-t^2) with s > t > 0 as positive integers, gcd(s, t) = 1 and s+t odd it is possible to generate all PPTs with 2^n-1 as the length of the odd leg providing that s = (x+y)/2 and t = (y-x)/2.
%C A297294 If 2^n-1 has d distinct prime factors (A046800(n)), then the set of pairs {x, y} such that x*y = 2^n-1, x < y and gcd(x, y) = 1 has a cardinality of 2^(d-1). This is because an integer m consisting of d distinct factors will have 2^d divisors and will generate pairs {x', y'} such that x'*y' = m, x' < y' and gcd(x', y') = 1 with a cardinality of 2^(d-1). Let m be the product of the distinct factor of 2^n-1 and r be the remainder consisting of the remaining repeated prime factors where m*r = 2^n-1. Then there has to be a 1 to 1 correspondence between the set of pairs {x', y'} created from the distinct prime factors of 2^n-1 and {x, y} created from all the prime factors of 2^n-1 whenever the repeated prime factors of r are combined with the distinct factors of m in the pairs {x, y} in order to preserve gcd(x, y) = 1.
%H A297294 Frank M Jackson, <a href="/A297294/b297294.txt">Table of n, a(n) for n = 1..928</a>
%F A297294 For n=1, a(n)=0 otherwise a(n)=2^(A046800(n)-1).
%e A297294 a(6)=2, because 2^6-1 = 63 gives pairs {1, 63}, {3, 21}, {7, 9} whose members when multiplied give 63. However, only two of these pairs are coprime and will generate PPTs.
%t A297294 pairs[n_] := Module[{m=2^n-1, lst=Divisors[2^n-1]}, Table[{lst[[l]], m/lst[[l]]}, {l, 1, Length[lst]/2}]]; Table[Length@Select[pairs[n], GCD@@#==1 &], {n, 1, 100}]
%t A297294 a[n_] := If[n==1, 0, 2^(Length@FactorInteger[2^n-1]-1)]; Array[a, 100]
%Y A297294 Cf. A001265, A046800.
%K A297294 nonn
%O A297294 1,4
%A A297294 _Frank M Jackson_, Jan 04 2018