A255572 -id:A255572 - cp's OEIS frontend

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

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.]

A260426 a(1) = 1, a(A206074(n)) = A014580(a(n)), a(A205783(1+n)) = A091242(a(n)), where A014580 [respectively A091242] give binary codes for irreducible [resp. reducible] polynomials over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 11, 6, 8, 12, 25, 9, 13, 17, 10, 14, 47, 18, 19, 34, 15, 20, 31, 24, 55, 16, 21, 62, 137, 26, 37, 27, 45, 22, 28, 42, 59, 33, 71, 23, 87, 29, 41, 79, 166, 35, 61, 49, 36, 58, 30, 38, 319, 54, 91, 76, 44, 89, 97, 32, 203, 108, 39, 53, 99, 200, 67, 46, 103, 78, 185, 64, 131, 48, 75, 40, 379, 50, 73, 373, 109, 70, 433, 113, 95, 57, 1123, 111, 143, 121

Offset: 1

Antti Karttunen, Jul 26 2015

Each term of A260427 resides in a separate infinite cycle. This follows because any polynomial with (coefficients 0 or 1) that is irreducible over GF(2) is also irreducible over Q, in other words, A014580 is a subset of A206074. [See Thomas Ordowski's Feb 21 2014 comment in A014580] and thus any term of A091242 in A206074 is trapped into a trajectory containing only terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..7968
Index entries for sequences that are permutations of the natural numbers

Inverse: A260425.
Related permutations: A246202, A245703, A260421, A260424.
Cf. A014580, A091242, A205783, A206074, A257000, A255572, A255574, A260427.
Differs from A245703 for the first time at n=25, where a(25)=55, while A245703(25)=16.

allocatemem(234567890);
vecsize = (2^24)-4;
uplim = 2^25;
v014580 = vector(vecsize); A014580 = n -> v014580[n];
v091242 = vector(vecsize); A091242 = n -> v091242[n];
v255574 = vector(vecsize); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA206074(n) = polisirreducible(Pol(binary(n)));
A257000 = n -> isA206074(n);
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); v255574[n] = v255574[n-1]+A257000(n); if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++); print(n);
A260426(n) = if(1==n, 1, if(isA206074(n), A014580(A260426(A255574(n))), A091242(A260426(A255572(n)))));
for(n=1, 7968, write("b260426.txt", n, " ", A260426(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A014580(a(A255574(n))), otherwise [when n is in A205783], a(n) = A091242(a(A255572(n))).
As a composition of related permutations:
a(n) = A246202(A260421(n)).
a(n) = A245703(A260424(n)).

A260421 a(1) = 1, a(A206074(n)) = 1 + (2a(n)), a(A205783(1+n)) = 2a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 29, 62, 24, 26, 9, 20, 61, 56, 16, 22, 63, 120, 25, 58, 124, 48, 52, 18, 27, 40, 122, 112, 21, 32, 57, 44, 126, 240, 17, 50, 116, 248, 96, 104, 23, 36, 121, 54, 80, 244, 59, 224, 125, 42, 64, 114, 88, 252, 49, 480, 53, 34, 19, 100, 41, 232, 496, 192, 123, 208, 113, 46, 33, 72, 45

Offset: 1

Antti Karttunen, Jul 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260422.
Related permutations: A246201, A246377, A260424, A260426.
Cf. A257000, A255572, A255574.

allocatemem(123456789);
uplim = 2^20;
v255574 = vector(uplim); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
isA206074(n) = polisirreducible(Pol(binary(n)));
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), v255574[n] = v255574[n-1]+isA206074(n); n++);
A260421(n) = if(1==n, 1, if(isA206074(n), 1 + 2*(A260421(A255574(n))), 2*(A260421(A255572(n)))));
for(n=1, 8192, write("b260421.txt", n, " ", A260421(n)));

If A257000(n) = 1 [when n is one of the terms of A206074] then a(n) = 1 + 2*a(A255574(n)), otherwise a(n) = 2*A260421(A255572(n)).
As a composition of related permutations:
a(n) = A246377(A260424(n)).
a(n) = A246201(A260426(n)).

A255573 a(n) = Number of terms of A205783 (including 1) in range 0 .. n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 14 2015

nonn

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

Essentially one more than A255572 (after the initial zero).

Cf. A062298, A205783, A255574.

A255573_write_bfile(up_to_n) = { my(n,a_n=0); for(n=0, up_to_n, if(((n > 0) && !polisirreducible(Pol(binary(n)))),a_n++); write("b255573.txt", n, " ", a_n)); };
A255573_write_bfile(8192);

a(n) = n - A255574(n).
Other identities and observations. For all n >= 1:
a(n) = 1 + A255572(n).
a(n) <= A062298(n).

A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 25, 26, 27, 31, 28, 37, 30, 32, 33, 34, 35, 41, 36, 44, 38, 43, 39, 47, 40, 46, 42, 53, 54, 45, 48, 49, 50, 59, 51, 61, 58, 52, 63, 67, 55, 71, 62, 56, 66, 57, 65, 73, 60, 79, 75, 83, 76, 89, 64, 68, 69, 109, 70, 97, 82, 101, 72, 103, 85, 81, 74, 127

Offset: 1

Antti Karttunen, Jul 25 2015

nonn

After 1, each term of A206075 resides in a separate infinite cycle. This follows because primes (A000040) is a subsequence of A206074 [see Thomas Ordowski's Feb 19 2014 comment in A206074] and thus each composite in A206074 is trapped into a trajectory containing only primes.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260423.
Related permutations: A245704, A246378, A260421, A260426.
Cf. A000040, A002808, A205783, A206074, A206075, A257000, A255572, A255574.

allocatemem(123456789);
default(primelimit,4294965247);
uplim = 2^20;
v255574 = vector(uplim); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
A257000(n) = polisirreducible(Pol(binary(n)));
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), v255574[n] = v255574[n-1]+A257000(n); n++);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
A260424(n) = if(1==n, 1, if(A257000(n), prime(A260424(A255574(n))), A002808(A260424(A255572(n)))));
for(n=1, 8192, write("b260424.txt", n, " ", A260424(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A000040(a(A255574(n))), otherwise [when n is in A205783], a(n) = A002808(a(A255572(n))).
As a composition of related permutations:
a(n) = A246378(A260421(n)).
a(n) = A245704(A260426(n)).

A283993 a(n) = number of reducible Stern polynomials in range B(1,x) .. B(n,x). (Polynomial B_1(x) = 1 is not included in the count).

Original entry on oeis.org

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, 43, 44, 44, 45, 46, 47, 47, 48, 48, 49, 50, 51, 51, 52, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 60, 61, 61, 62, 63, 64, 64

Offset: 1

Antti Karttunen, Mar 20 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A125184, A260443, A283992, A283994.

Differs from A255572 for the first time at n=65, where a(65) = 43, while A255572(65) = 44.

(define (A283993 n) (- n (A283992 n) 1))

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

cp's OEIS Frontend

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

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A260421 a(1) = 1, a(A206074(n)) = 1 + (2*a(n)), a(A205783(1+n)) = 2*a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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A255573 a(n) = Number of terms of A205783 (including 1) in range 0 .. n.

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Formula

A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A283993 a(n) = number of reducible Stern polynomials in range B(1,x) .. B(n,x). (Polynomial B_1(x) = 1 is not included in the count).

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Author

Keywords

Links

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Programs

Scheme

Formula

A260421 a(1) = 1, a(A206074(n)) = 1 + (2a(n)), a(A205783(1+n)) = 2a(n), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.