A186891 Numbers n such that the Stern polynomial B(n,x) is irreducible.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 157, 161, 163, 167, 169, 173, 175, 179, 181, 185, 191, 193, 197, 199

Offset: 1

Author

T. D. Noe, Feb 28 2011

Keywords

• nonn

Comments

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.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Maciej Ulas and Oliwia Ulas, On certain arithmetic properties of Stern polynomials, arXiv:1102.5109 [math.CO], 2011.

Crossrefs

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

Cf. A186892 (subsequence of nonprime terms).

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

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

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

Cf. also A186886, A277190, A277318, A283992.

Programs

Mathematica

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]]]];
selQ[n_] := IrreduciblePolynomialQ[ps[n]];
Join[{1}, Select[Range[200], selQ]] (* Jean-François Alcover, Nov 02 2018, translated from PARI *)

PARI

ps(n)=if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)))
is(n)=polisirreducible(ps(n)) \\ Charles R Greathouse IV, Apr 07 2015

Formula

From Antti Karttunen, Mar 21 2017: (Start)

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

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

(End)