%I #9 May 22 2024 15:12:27

%S 1,341,671,1891,2117,3277,4033,5461,8249,12557,13021,14531,19171,

%T 24811,31609,32777,33437,40951,46139,48929,49981,50737,73279,80581,

%U 84169,100253,116143,130289,135923,136271,149437,175577,179783,194417,252361,272491,342151,343027,376169,390641

%N Squarefree terms of A372894 whose prime factors are neither elite (A102742) nor anti-elite (A128852).

%C By construction, A372894 is the disjoint union of the two following sets of numbers: (a) products of a square, some distinct anti-elite primes, an even number of elite-primes and a term here; (b) products of a square, some distinct anti-elite primes, an odd number of elite-primes and a term in A372895.

%o (PARI) isA372896(n) = {

%o if(n%2 && issquarefree(n) && isA372894(n), if(n==1, return(1)); my(f = factor(n)~[1,]); \\ See A372894 for its program

%o for(i=1, #f, my(p=f[i], d = znorder(Mod(2, p)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint)), flag = 0); \\ To check if p = f[i] is an elite prime or an anti-elite prime, it suffices to check (2^2^i + 1) modulo p for StartPoint <= i <= StartPoint + LengthTest - 1; see A129802 or A372894

%o for(j = StartPoint+1, StartPoint + LengthTest - 1, if(issquare(Mod(2, p)^2^j + 1) != issquare(Mod(2, p)^2^StartPoint + 1), flag = 1; break())); if(flag == 0, return(0))); 1, 0)

%o }

%Y Cf. A372894, A102742, A128852, A129802, A372895.

%K nonn

%O 1,2

%A _Jianing Song_, May 15 2024