A305300 Ordinal transform of A305430, the smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over Q.

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], a(n) = 1, otherwise a(n) = 1 + n - A305429(n).

Mathematica

binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];
ip[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];
A305430[n_] := Module[{k = n + 1}, While[ip[k] == 0, k++]; k];
b[_] = 0;
a[n_] := a[n] = With[{t = A305430[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

Prog

(PARI)
A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3, 1, my(k=n-1); while(k>1 && !A257000(k), k--); (k));
A305300(n) = if((1==n)||(1==A257000(n)), 1, 1+(n-A305429(n)));
(PARI)
up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };
A305430(n) = { my(k=1+n); while(!A257000(k), k++); (k); };
v305300 = ordinal_transform(vector(up_to, n, A305430(n)));
A305300(n) = v305300[n];

Crossrefs

Cf. A206074 (gives the positions of other 1's after the initial one).

Cf. A257000, A305429, A305430, A305437, A305438.

Cf. also A175851.

Sequence in context: A161296 A161271 A160975 * A330241 A175851 A049711

Adjacent sequences: A305297 A305298 A305299 * A305301 A305302 A305303

Keyword

nonn

Author

Antti Karttunen, Jun 09 2018

Status

approved