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Nontrivial odd solutions n to P(n) == 1+r*Sum_{i=1..d(n)} x^i mod m where P(n) is the n-th Stern polynomial, d(n) is the degree of P(n), r=0, m=3.
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%I #10 Sep 29 2019 02:47:53

%S 19,181,29899,40123,44659,72361,87211,183439,373465,2965429,5073589,

%T 17484211,44733781,165459277,1381288843,2572135705,2833893901

%N Nontrivial odd solutions n to P(n) == 1+r*Sum_{i=1..d(n)} x^i mod m where P(n) is the n-th Stern polynomial, d(n) is the degree of P(n), r=0, m=3.

%C Here, nontrivial means solutions neither of the form 2^(k+1)-1 nor of the form 2^(k+2)-3.

%C Sequence is infinite as it includes the infinite subsequence h(n) = 2*(2^(2*n)-1)*(2^(2*n+1) + 1)/3 + 1. See link.

%H Maciej Ulas, <a href="https://arxiv.org/abs/1909.10844">Strong arithmetic property of certain Stern polynomials</a>, arXiv:1909.10844 [math.NT], 2019. See Table 2 p. 11.

%o (PARI) pol(n) = {if (n<2, return (n)); if (n%2, pol((n+1)/2) + pol((n-1)/2), x*pol(n/2));} \\ A125184

%o ispow2(n) = if ((n==1) || (n==2), return (1)); my(p); isprimepower(n,&p) && (p==2);

%o istrivial(n) = ispow2(n+1) || ispow2(n+3);

%o isokrm(n, r, m) = {if ((n%2) && !istrivial(n), my(p=pol(n), d=poldegree(p)); Mod(p, m) == Mod(1+r*sum(i=1, d, x^i), m););}

%o lista(nn) = forstep(n=1, nn, 2, if (isokrm(n, 0, 3), print1(n, ", ")));

%Y Cf. A125184 (Stern polynomials), A327849, A327850.

%K nonn,more

%O 1,1

%A _Michel Marcus_, Sep 28 2019