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A354381 Primitive elements in A354379, being those not divisible by any previous term.

%I A354381 #26 May 11 2023 09:15:31
%S A354381 25,65,85,89,109,145,149,169,173,185,205,221,229,233,265,289,293,305,
%T A354381 313,349,353,365,377,409,421,433,449,461,481,485,493,505,509,533,565,
%U A354381 601,613,629,641,653,677,685,689,697,709,757,761,769,773,785,793,797,821,829,841,857,877,881,901,905
%N A354381 Primitive elements in A354379, being those not divisible by any previous term.
%H A354381 Robert Israel, <a href="/A354381/b354381.txt">Table of n, a(n) for n = 1..10000</a>
%e A354381 The primitive Pythagorean triple (39, 80, 89) has all its terms in A009003, and 89 is not divisible by any previous term. Hence 89 is in sequence.
%p A354381 ishyp:= proc(n) local s; ormap(s -> s mod 4 = 1, numtheory:-factorset(n)) end proc:
%p A354381 filter:= proc(n) local s;
%p A354381   ormap(s -> ishyp(subs(s,x)) and ishyp(subs(s,y)), [isolve(x^2+y^2=n^2)])
%p A354381 end proc:
%p A354381 R:= []: count:= 0:
%p A354381 for n from 1 while count < 100 do
%p A354381   if ormap(t -> n mod t = 0, R) then next fi;
%p A354381   if filter(n) then R:= [op(R),n]; count:= count+1; fi
%p A354381 od:
%p A354381 R; # _Robert Israel_, Jan 10 2023
%t A354381 ishyp[n_] := AnyTrue[ FactorInteger[n][[All, 1]], Mod[#, 4] == 1 &] ;
%t A354381 filter[n_] := AnyTrue[Solve[x^2 + y^2 == n^2, Integers], ishyp[x /. #] && ishyp[y /. #] &];
%t A354381 R = {}; count = 0;
%t A354381 For[n = 1, count < 100, n++, If[AllTrue[R, Mod[n, #] != 0&], If[filter[n], AppendTo[R, n]; count++]]];
%t A354381 R (* _Jean-François Alcover_, May 11 2023, after _Robert Israel_ *)
%Y A354381 Cf. A009003, A008846, A020882, A004613, A354379.
%K A354381 nonn
%O A354381 1,1
%A A354381 _Lamine Ngom_, May 24 2022
%E A354381 Corrected by _Robert Israel_, Jan 10 2023