A344186 a(n) = A344142(n) - 2^n.

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, 9, 141, 1025, 129, 5, 513, 83, 99, 17, 57, 9, 129, 89, 33, 27, 3, 33, 45, 513, 29, 75, 9, 71, 513, 129, 149, 17, 524289, 149, 3, 39, 536870913, 3, 27, 262145, 9, 39, 513, 101, 43

Offset

1,2

Comments

A more intuitive version of A344142.

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.

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.

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.

Links

Jianing Song, Table of n, a(n) for n = 1..1000

Example

See A344142.

Prog

(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.

Crossrefs

Cf. A014580, A344142, A344185, A344177.

Sequence in context: A125958 A247244 A344185 * A132448 A132450 A132424

Adjacent sequences: A344183 A344184 A344185 * A344187 A344188 A344189

Keyword

nonn

Author

Jianing Song, May 11 2021

Status

approved