A305430 -id:A305430 - cp's OEIS frontend

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

Antti Karttunen, Jun 09 2018

nonn

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

Cf. A206074 (gives the positions of other 1's after the initial one).
Cf. A257000, A305429, A305430, A305437, A305438.
Cf. also A175851.

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 *)

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)));

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];

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).

A305420 Smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X].

Original entry on oeis.org

2, 3, 7, 7, 7, 7, 11, 11, 11, 11, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 55, 55, 55, 55, 55, 55, 55, 55, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 73, 73, 73, 73, 73, 73, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 91

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

a(n) is the smallest term of A014580 greater than n.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A014580, A091225, A091228, A305419, A305421.

Cf. also A151800, A305430.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };

For n >= 1, a(n) = A091228(1+n).

A305429 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over Q, with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 25, 25, 25, 25, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 55, 55, 55, 55, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 69, 69, 71, 71, 73, 73, 73, 73, 77, 77, 79, 79, 81, 81, 83, 83, 83, 83, 87, 87

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

For n >= 3, a(n) is the largest term of A206074 less than n.

Antti Karttunen, Table of n, a(n) for n = 1..21845

Cf. A206074, A257000, A305430.

Cf. also A151799 (A136548), A305419.

A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3,1, my(k=n-1); while(k>1 && !A257000(k),k--); (k));

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

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A305420 Smallest k > n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X].

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A305429 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over Q, with a(1) = a(2) = 1.

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PARI