A255573 -id:A255573 - cp's OEIS frontend

A205783 Complement of A206074, a coding of reducible polynomials over Q (with coefficients 0 or 1).

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Clark Kimberling, Feb 03 2012

nonn

Reducibility here refers to the field of rational numbers.

Except for its initial 3, is A039004 a subsequence of A205783?

			The reducible polynomials matching the first four terms:
1 = 1(base 2) matches 1
4 = 100(base 2) matches x^2
6 = 110(base 2) matches x^2 + x
8 = 1000(base 2) matches x^3
9 = 1001(base 2) matches x^3 + 1

Antti Karttunen, Table of n, a(n) for n = 1..21951

Cf. A206074 (complement), A255573 (left inverse).
After 1 a subsequence of A091212 (69 is the first term missing from here).
Cf. also permutations A260421 - A260426.

t = Table[IntegerDigits[n, 2], {n, 1, 850}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[u, n]], {n, 300}];
u                          (* A206074 *)
Complement[Range[200], u]  (* A205783 *)
b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
Table[b[n], {n, 1, 40}]    (* A206073 *)

isA205783(n) = ((n > 0) && !polisirreducible(Pol(binary(n))));
n = 0; i = 0; while(n < 32768, n++; if(isA205783(n), i++; write("b205783.txt", i, " ", n)));
\\ Antti Karttunen, Jul 28 2015 after Joerg Arndt's code for A206074.

Other identities and observations. For all n >= 1:

A255573(a(n)) = n.

A255574 a(n) = Number of terms of A206074 in range 0 .. n.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 27, 27

Offset: 0

Antti Karttunen, May 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Partial sums of A257000.

Cf. A000720, A014580, A091226, A206074, A255572, A255573.

binPol[n_, x_] := With[{bb = IntegerDigits[n, 2]}, bb.x^Range[Length[bb]-1, 0, -1]];
b[n_] := If[IrreduciblePolynomialQ[binPol[n, x]], 1, 0];
b /@ Range[0, 128] // Accumulate (* Jean-François Alcover, Dec 20 2021 *)

isA206074(n) = polisirreducible(Pol(binary(n)));
A255574_write_bfile(up_to_n) = { my(n,a_n=0); for(n=0, up_to_n, if(isA206074(n),a_n++); write("b255574.txt", n, " ", a_n)); };
A255574_write_bfile(65537);

(definec (A255574 n) (if (zero? n) n (+ (A257000 n) (A255574 (- n 1)))))

a(0) = 0; for n >= 1, a(n) = A257000(n) + a(n-1).
Other identities and observations.
For all n >= 0:
a(n) = n - A255573(n).
For all n >= 1:
a(A206074(n)) = n. [This sequence works as a left inverse for injection A206074.]
a(n) >= A000720(n). [Because primes is a subsequence of A206074.]
a(n) >= A091226(n). [Because A014580 is a subsequence of A206074.]

A255572 a(n) = Number of terms larger than one in range 0 .. n of A205783.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 40, 41, 42, 43, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 50, 51, 51, 52, 52, 53, 53

Offset: 0

Antti Karttunen, May 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Essentially one less than A255573 (after the initial zero).

Cf. A065855, A091245, A205783, A255574.

A255572_write_bfile(up_to_n) = { my(n,a_n=0); for(n=0, up_to_n, if(((n > 1) && !polisirreducible(Pol(binary(n)))),a_n++); write("b255572.txt", n, " ", a_n)); };
A255572_write_bfile(8192);

Other identities and observations. For all n >= 1:
a(n) = A255573(n) - 1.
a(n) <= A065855(n).
a(n) <= A091245(n).

cp's OEIS Frontend

A205783 Complement of A206074, a coding of reducible polynomials over Q (with coefficients 0 or 1).

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A255574 a(n) = Number of terms of A206074 in range 0 .. n.

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A255572 a(n) = Number of terms larger than one in range 0 .. n of A205783.

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