cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 23 results. Next

A091225 Characteristic function of A014580: 1 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

a(n) = A010051(A091203(n)) = A010051(A091205(n)). Partial sums give A091226. Cf. A091227. Complementary to A091247.

Programs

  • PARI
    a(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ Michel Marcus, Nov 11 2017

Extensions

Data section extended up to a(120) by Antti Karttunen, Jan 01 2023

A091205 Factorization and index-recursion preserving isomorphism from binary codes of GF(2) polynomials to integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 23, 10, 27, 16, 81, 30, 13, 36, 25, 14, 69, 24, 11, 46, 45, 20, 21, 54, 19, 32, 57, 162, 115, 60, 47, 26, 63, 72, 61, 50, 33, 28, 135, 138, 17, 48, 35, 22, 243, 92, 39, 90, 37, 40, 207, 42, 83, 108, 29, 38, 75, 64, 225, 114, 103
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

This "deeply multiplicative" bijection is one of the deep variants of A091203 which satisfy most of the same identities as the latter, but it additionally preserves also the structures where we recurse on irreducible polynomial's A014580-index. E.g., we have: A091238(n) = A061775(a(n)). The reason this holds is that when the permutation is restricted to the binary codes for irreducible polynomials over GF(2) (A014580), it induces itself: a(n) = A049084(a(A014580(n))).
On the other hand, when this permutation is restricted to the union of {1} and reducible polynomials over GF(2) (A091242), permutation A245813 is induced.

Links

Crossrefs

Inverse: A091204.
Similar or related permutations: A091203, A106443, A106445, A106447, A235042, A245704, A245813, A245821.
Cf. A000040, A007097, A010051, A014580, A049084, A061775, A091238, A091225, A091226, A091230, A091242.

Programs

  • PARI
    allocatemem(123456789);
v091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++)
A091226(n) = v091226[n];
A091205(n) = if(n<=1,n,if(isA014580(n),prime(A091205(A091226(n))),{my(irfs,t); irfs=subst(lift(factor(Mod(1,2)*Pol(binary(n)))),x,2); irfs[,1]=apply(t->A091205(t),irfs[,1]); factorback(irfs)}));
for(n=0, 8192, write("b091205.txt", n, " ", A091205(n)));
\\ Antti Karttunen, Aug 16 2014

Formula

a(0)=0, a(1)=1. For n that is coding an irreducible polynomial, that is if n = A014580(i), we have a(n) = A000040(a(i)) and for reducible polynomials a(ir_i X ir_j X ...) = a(ir_i) * a(ir_j) * ..., where ir_i = A014580(i), X stands for carryless multiplication of polynomials over GF(2) (A048720) and * for the ordinary multiplication of integers (A004247).
As a composition of related permutations:
a(n) = A245821(A245704(n)).
Other identities.
For all n >= 0, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A245704 has the same property.]
For all n >= 1, the following holds:
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, bijectively to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers, in some order. The permutations A091203, A106443, A106445, A106447, A235042 and A245704 have the same property.]

Extensions

Name changed by Antti Karttunen, Aug 16 2014

A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 25, 14, 18, 19, 22, 26, 33, 38, 24, 11, 28, 30, 34, 39, 49, 23, 55, 36, 20, 42, 45, 37, 50, 56, 69, 47, 35, 77, 52, 32, 60, 17, 64, 54, 70, 78, 94, 66, 51, 29, 105, 74, 48, 41, 84, 53, 27, 88, 76, 95, 106, 73, 125, 91, 72, 44, 140, 97, 100, 68, 58, 115, 75, 40
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Comments

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Links

Crossrefs

Inverse: A245703.
Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

Programs

Formula

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).
As a composition of related permutations:
a(n) = A227413(A245701(n)).
a(n) = A245822(A091205(n)).
Other identities. For all n >= 1, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

A091227 Inverse function of A014580: position in A014580 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 10, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 18, 0, 0, 0, 0, 0, 19, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

Inverse of A014580. a(n) = A049084(A091203(n)).

Formula

a(n) = A091225(n) * A091226(n).

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2*a(n), a(A091242(n)) = 2*a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 8, 7, 11, 19, 6, 17, 10, 15, 23, 39, 13, 35, 18, 21, 31, 47, 79, 27, 16, 71, 37, 43, 63, 95, 14, 159, 55, 33, 143, 75, 22, 87, 127, 191, 38, 29, 319, 111, 67, 287, 12, 151, 45, 175, 255, 383, 77, 59, 34, 639, 223, 135, 20, 575, 30, 25, 303, 91, 351, 511, 46, 767, 155, 119, 69, 1279, 78, 447, 271, 41, 1151, 61, 51
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Links

Crossrefs

Inverse: A245702.
Cf. A000035, A014580, A091242, A091225, A091226, A091245.
Similar entanglement permutations: A135141, A193231, A237427, A243287, A245703, A245704.

Programs

Formula

a(1) = 1, and for n > 1, if n is in A014580, a(n) = 2*a(A091226(n)), otherwise a(n) = 1 + 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A135141(A245704(n)).
Other identities:
For all n >= 1, 1 - A000035(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (= A014580) to even numbers and the corresponding representations of reducible polynomials to odd numbers].

A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 8, 12, 11, 15, 23, 81, 18, 10, 17, 30, 13, 162, 27, 36, 19, 24, 16, 25, 38, 46, 37, 45, 31, 135, 14, 20, 50, 57, 47, 69, 21, 55, 83, 115, 419, 87, 60, 210, 61, 42, 54, 26, 90, 28, 29, 35, 32, 63, 171, 52, 59, 138, 113, 180, 111, 48, 88, 39, 41, 621, 72, 22, 953, 230, 103, 207, 126, 64, 33, 243
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Links

Crossrefs

Inverse: A245822.
Other related permutations: A091205, A245703, A245815.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

Programs

  • PARI
    allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
for(n=1, 10001, write("b245821.txt", n, " ", A245821(n)));
  • Scheme
    (define (A245821 n) (A091205 (A245703 n)))

Formula

a(n) = A091205(A245703(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245815(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A062692 Number of irreducible polynomials over F_2 of degree at most n.

Original entry on oeis.org

2, 3, 5, 8, 14, 23, 41, 71, 127, 226, 412, 747, 1377, 2538, 4720, 8800, 16510, 31042, 58636, 111013, 210871, 401428, 766150, 1465020, 2807196, 5387991, 10358999, 19945394, 38458184, 74248451, 143522117, 277737797, 538038783, 1043325198
Offset: 1

Views

Author

Gary L Mullen (mullen(AT)math.psu.edu), Jul 04 2001

Keywords

Comments

Number of binary pre-necklaces of length n. - Joerg Arndt, Jul 20 2013

Links

Crossrefs

Partial sums of A001037.
Cf. A014580, A091226, A091231.
Equals A001036 + 1.
Column k=2 of A143328. - Alois P. Heinz, Jul 20 2013

Programs

  • Maple
    with(numtheory):for n from 1 to 113 do sum3 := 0:for m from 1 to n do sum2 := 0:a := divisors(m):for h from 1 to nops(a) do sum2 := sum2+mobius(a[h])*2^(m/a[h]):end do:sum3 := sum3+sum2/m:end do:s[n] := sum3:end do:q := seq(s[j],j=1..113);
  • Mathematica
    a[n_] := Sum[1/m DivisorSum[m, MoebiusMu[#]*2^(m/#)&], {m, 1, n}]; Array[a, 34] (* Jean-François Alcover, Dec 07 2015 *)
f[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#) &] / n; Accumulate[Array[f, 30]] (* Amiram Eldar, Aug 24 2023 *)
  • PARI
    a(n)=sum(m=1,n, 1/m* sumdiv(m, d, moebius(d)*2^(m/d) ) ); /* Joerg Arndt, Jul 04 2011 */

Formula

a(n) = Sum_{m=1..n} (1/m)*Sum_{d | m } mu(d)*2^{m/d}.
a(n) = A091226(2^(n+1)).
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k)*log(1/(1 - 2*x^k))/k. - Ilya Gutkovskiy, Nov 11 2019

Extensions

More terms from Sascha Kurz, Mar 25 2002

A245822 Permutation of natural numbers: a(n) = A245704(A091204(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 9, 6, 16, 11, 10, 19, 33, 12, 25, 17, 15, 23, 34, 39, 70, 13, 24, 26, 50, 21, 52, 53, 18, 31, 55, 77, 93, 54, 22, 29, 27, 66, 105, 67, 48, 137, 156, 30, 28, 37, 64, 91, 35, 85, 58, 97, 49, 40, 98, 36, 135, 59, 45, 47, 261, 56, 76, 92, 122, 83, 374, 38, 102, 139, 69, 167, 130, 88, 203, 351, 212, 349, 235, 14
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2014

Keywords

Links

Crossrefs

Inverse: A245821.
Other related permutations: A091204, A245704, A245816.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

Programs

Formula

a(n) = A245704(A091204(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245816(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2*a(n))+1, a(A091242(n)) = 2*a(n), where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 3, 7, 2, 6, 14, 15, 4, 12, 28, 5, 30, 13, 8, 24, 56, 10, 60, 29, 26, 16, 48, 112, 20, 31, 120, 58, 52, 32, 96, 9, 224, 40, 62, 240, 116, 25, 104, 64, 192, 57, 18, 448, 80, 124, 480, 11, 232, 50, 208, 128, 384, 114, 36, 61, 896, 160, 248, 27, 960, 17, 22, 464, 100, 416, 256, 49, 768, 228, 72, 122, 1792, 113, 320, 496, 54, 1920, 34, 44
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2014

Keywords

Comments

Because 2 is the only even term in A014580, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
Note that for any value k in A246156, "Odd reducible polynomials over GF(2)": 5, 9, 15, 17, 21, 23, ..., a(k) will be even, and apart from 2, all other even numbers are mapped to some even number, so all those terms reside in infinite cycles. Furthermore, apart from 5 and 15, all of them reside in separate cycles. The infinite cycle containing 5 and 15 goes as: ..., 47, 11, 5, 6, 14, 8, 4, 2, 3, 7, 15, 24, 20, 26, 120, 7680, ... and it is only because a(2) = 3, that it can temporarily switch back from even terms to odd terms, until after a(15) = 24 it is finally doomed to the eternal evenness.
(Compare also to the comments given at A246161).

Links

Crossrefs

Inverse: A246202.
Similar or related permutations: A245701, A246161, A006068, A054429, A193231, A246163, A246203, A237427.
Cf. A091225, A091226, A091245.

Formula

a(1) = 1, and for n > 1, if A091225(n) = 1 [i.e. when n is in A014580], a(n) = 1 + (2*a(A091226(n))), otherwise a(n) = 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A054429(A245701(n)).
a(n) = A006068(A246161(n)).
a(n) = A193231(A246163(n)).
a(n) = A246203(A193231(n)).
Other identities:
For all n > 1, A000035(a(n)) = A091225(n). [After 1 maps binary representations of reducible GF(2) polynomials to even numbers and the corresponding representations of irreducible polynomials to odd numbers, in some order. A246203 has the same property].

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

Views

Author

Antti Karttunen, May 14 2015

Keywords

Links

Crossrefs

Partial sums of A257000.
Cf. A000720, A014580, A091226, A206074, A255572, A255573.

Programs

  • Mathematica
    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 *)
  • PARI
    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);
  • Scheme
    (definec (A255574 n) (if (zero? n) n (+ (A257000 n) (A255574 (- n 1)))))

Formula

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.]
Showing 1-10 of 23 results. Next

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