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Numbers n such that the Stern polynomial B(n,x) is irreducible.
30

%I #30 Nov 03 2018 12:15:26

%S 1,2,3,5,7,11,13,17,19,23,25,29,31,37,41,43,47,53,55,59,61,65,67,71,

%T 73,77,79,83,89,91,95,97,101,103,107,109,113,115,121,125,127,131,133,

%U 137,139,143,145,149,151,157,161,163,167,169,173,175,179,181,185,191,193,197,199

%N Numbers n such that the Stern polynomial B(n,x) is irreducible.

%C Ulas and Ulas conjecture that all primes are here. The nonprime n are in A186892. See A186886 for the least number having n prime factors.

%H Charles R Greathouse IV, <a href="/A186891/b186891.txt">Table of n, a(n) for n = 1..10000</a>

%H Maciej Ulas and Oliwia Ulas, <a href="http://arxiv.org/abs/1102.5109">On certain arithmetic properties of Stern polynomials</a>, arXiv:1102.5109 [math.CO], 2011.

%F From _Antti Karttunen_, Mar 21 2017: (Start)

%F A283992(a(1+n)) = n.

%F A260443(a(1+n)) = A277318(n).

%F (End)

%t ps[n_] := ps[n] = If[n<2, n, If[OddQ[n], ps[Quotient[n, 2]] + ps[Quotient[n, 2] + 1], x ps[Quotient[n, 2]]]];

%t selQ[n_] := IrreduciblePolynomialQ[ps[n]];

%t Join[{1}, Select[Range[200], selQ]] (* _Jean-François Alcover_, Nov 02 2018, translated from PARI *)

%o (PARI) ps(n)=if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)))

%o is(n)=polisirreducible(ps(n)) \\ _Charles R Greathouse IV_, Apr 07 2015

%Y Cf. A057526 (degree of Stern polynomials), A125184, A260443 (Stern polynomials).

%Y Cf. A186892 (subsequence of nonprime terms).

%Y Cf. A186893 (subsequence for self-reciprocal polynomials).

%Y Positions of 0 and 1's in A277013, Positions of 1 and 2's in A284011.

%Y Cf. A283991 (characteristic function for terms > 1).

%Y Cf. also A186886, A277190, A277318, A283992.

%K nonn

%O 1,2

%A _T. D. Noe_, Feb 28 2011