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.

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A305418 Permutation of nonnegative integers: a(1) = 0, a(2n) = 1 + 2*a(n), a(2n+1) = 2*a(A305422(2n+1)).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 10, 13, 8, 11, 16, 9, 14, 15, 30, 21, 32, 27, 12, 17, 34, 23, 64, 33, 22, 19, 18, 29, 128, 31, 258, 61, 36, 43, 256, 65, 38, 55, 512, 25, 130, 35, 46, 69, 1024, 47, 20, 129, 62, 67, 66, 45, 2048, 39, 70, 37, 4096, 59, 8192, 257, 26, 63, 54, 517, 16384, 123, 24, 73, 16386, 87, 32768, 513, 142, 131, 8194, 77, 132, 111, 48, 1025, 42, 51
Offset: 1

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Author

Antti Karttunen, Jun 10 2018

Keywords

Comments

This is GF(2)[X] analog of A156552. Note the indexing: the domain starts from 1, while the range includes also zero.

Links

Crossrefs

Cf. A305417 (inverse).
Cf. A305422.
Cf. also A091203, A156552, A052331, A305428.

Programs

  • PARI
    A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305418(n) = if(1==n,(n-1),if(!(n%2),1+(2*(A305418(n/2))),2*A305418(A305422(n))));

Formula

a(1) = 0, a(2n) = 1 + 2*a(n), a(2n+1) = 2*a(A305422(2n+1)).
a(n) = A054429(A305428(n)).
For all n >= 1:
A000120(a(n)) = A091222(n).
A069010(a(n)) = A091221(n).
A106737(a(n)) = A091220(n).
A132971(a(n)) = A091219(n).
A085357(a(n)) = A304109(n).

A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 3, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 3, 4, 3, 9, 2, 4, 3, 8, 2, 6, 2, 6, 6, 4, 2, 10, 3, 4, 4, 6, 2, 8, 2, 8, 3, 4, 2, 12, 2, 4, 6, 7, 3, 6, 2, 6, 2, 6, 2, 12, 2, 4, 5, 6, 2, 6, 2, 10, 2, 4, 2, 9, 4, 4, 2, 8, 2, 12, 2, 6, 3, 4, 4, 12, 2, 6, 4, 6, 2, 8, 2, 8, 5
Offset: 1

Views

Author

Antti Karttunen, May 11 2019

Keywords

Comments

a(n) is the number of divisors d of n such that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), that polynomial is a divisor of the (0,1)-polynomial similarly converted from n, when the polynomial division is done over field GF(2).

Examples

			39 = 3*13 has four divisors 1, 3, 13, 39, of which all other divisors except 13 are counted because we have A048720(1,39) = A048720(39,1) = A048720(3,29) = 39, but A048720(13,u) is not equal to 39 for any u, thus a(39) = 3. See also the example in A325563.

Links

Crossrefs

Cf. A000005, A048720, A091220, A325559 (positions of 2's), A325563, A325565.

Programs

  • PARI
    A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };

Formula

For all n, A325565(n) <= a(n) <= min(A000005(n), A091220(n)).

A178909 Indices of perfect polynomials over GF(2).

Original entry on oeis.org

1, 6, 36, 54, 120, 2470, 2640, 3144, 3780, 32640, 41280, 52632, 67184, 1098176, 1157904, 2147450880
Offset: 1

Views

Author

Franklin T. Adams-Watters, Jun 22 2010

Keywords

Comments

Numbers k such that k = A178908(k); sum of divisors of k-th GF(2) polynomial is the polynomial itself.
a(17) > 5*10^9. - Amiram Eldar, Oct 28 2019

Crossrefs

Cf. A178908, A091220, A000396, A178911.

Programs

  • PARI
    isok(n) = my(s = vecsum(divisors(Mod(1,2)*Pol(binary(n))))); subst(lift(s), x, 2) == n; \\ Michel Marcus, Jan 13 2019

Extensions

a(14)-a(15) from Amiram Eldar, Jan 13 2019
a(16) from Amiram Eldar, Oct 28 2019
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