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A375204 Record values in A375202.

%I A375204 #25 Oct 17 2024 17:47:56
%S A375204 0,1,2,3,4,5,6,8,9,10,12,16,18,20,24,32,36,40,48,64,72,80,96,128,144,
%T A375204 160,192,256,288,320,384,512,576,640,768,1024,1152,1280,1536,2048,
%U A375204 2304,2560,3072,4096,4608,5120,6144,8192,9216,10240,12288,16384
%N A375204 Record values in A375202.
%C A375204 Numbers k such that k = A375202(m) for some m such that A375202(j) < k for all j < m.
%C A375204 Conjectures: All powers of 2 (A000079), 3*(powers of 2) (A007283) and 5*(powers of 2) (A020714) are terms. All terms are 5-smooth (A051037). - _Chai Wah Wu_, Oct 16 2024
%H A375204 Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/4985056/large-smallest-square-in-a-sum-of-three-squares#4985056">Large smallest square in a sum of three squares?</a>
%F A375204 a(n) = A375292(A375203(n)).
%e A375204 a(3) = 2 because A375202(12) = 2 and A375202(j) <= 1 for j < 12.
%p A375204 f:= proc(n) local q, x, y, z;
%p A375204   if n/4^padic:-ordp(n, 4) mod 8 = 7 then return -1 fi;
%p A375204   for x from 0 while 3*x^2 <= n do
%p A375204     if [isolve(y^2 + z^2 = n - x^2)] <> [] then return x fi
%p A375204   od;
%p A375204 end proc:
%p A375204 V:= NULL:count:= 0: m:= -1;
%p A375204 for i from 0 while count < 39 do
%p A375204   v:= f(i);
%p A375204   if v > m then
%p A375204     V:= V, v; m:= v; count:=count+1
%p A375204   fi
%p A375204 od:
%p A375204 V;
%o A375204 (Python)
%o A375204 from itertools import count, islice
%o A375204 from math import isqrt
%o A375204 from sympy import factorint
%o A375204 def A375204_gen(): # generator of terms
%o A375204     c = -1
%o A375204     for n in count(0):
%o A375204         v = (~n & n-1).bit_length()
%o A375204         if v&1 or n>>v&7!=7:
%o A375204             a = next(x for x in range(isqrt(n//3)+1) if not any(e&1 and p&3==3 for p, e in factorint(n-x**2).items()))
%o A375204             if a>c:
%o A375204                 yield a
%o A375204                 c = a
%o A375204 A375204_list = list(islice(A375204_gen(),20)) # _Chai Wah Wu_, Oct 16 2024
%Y A375204 Cf. A000079, A007283, A020714, A051037, A375202, A375203.
%K A375204 nonn,more
%O A375204 1,3
%A A375204 _Robert Israel_, Oct 15 2024
%E A375204 a(35)-a(48) from _Chai Wah Wu_, Oct 16 2024
%E A375204 a(49)-a(52) from _Chai Wah Wu_, Oct 17 2024