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%I #18 May 13 2021 06:04:13
%S 0,3,3,3,5,3,3,27,3,9,5,9,27,33,3,43,9,9,39,9,5,3,33,27,9,27,39,3,5,3,
%T 9,141,1025,129,5,513,83,99,17,57,9,129,89,33,27,3,33,45,513,29,75,9,
%U 71,513,129,149,17,524289,149,3,39,536870913,3,27,262145,9,39,513,101,43
%N a(n) = A344142(n) - 2^n.
%C A more intuitive version of A344142.
%C In A057496 it is stated that if x^n + x^3 + x^2 + x + 1 is irreducible, then so is x^n + x^3 + 1. It follows that no term can be equal to 15.
%C It is conjectured that an irreducible polynomial of degree n with 5 terms exists for every n. It follows from the conjecture that for n >= 2, a(n) is of the form 2^k + 1 or an odd number with Hamming weight 4.
%C It is conjectured that no term can be of the form P_m(2^k), where P_m(x) = Product_{i>=0} (1 + x^(2^(d_i)))^(c_i) if the binary representation of m is m = Sum_{i>=0} c_i * 2^(d_i), k is an odd number. See my conjecture in A344177.
%H Jianing Song, <a href="/A344186/b344186.txt">Table of n, a(n) for n = 1..1000</a>
%e See A344142.
%o (PARI) A344186(n) = if(n==1, 0, for(k=1, n-1, if(polisirreducible(Mod(x^n+x^k+1, 2)), return(2^k+1))); for(a=3, n-1, for(b=2, a-1, for(c=1, b-1, if(polisirreducible(Mod(x^n+x^a+x^b+x^c+1, 2)), return(2^a+2^b+2^c+1)))))) \\ Assuming that an irreducible polynomial of degree n with at most 5 terms exists for every n.
%Y Cf. A014580, A344142, A344185, A344177.
%K nonn
%O 1,2
%A _Jianing Song_, May 11 2021
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