A091205 -id:A091205 - cp's OEIS frontend

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.

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

Antti Karttunen, Aug 02 2014

nonn

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.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

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.

allocatemem(123456789);
default(primelimit, 2^22)
A091226 = vector(2^22);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));

;; With memoization-macro definec.
(definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))

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].

A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185

Offset: 0

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

nonn

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).

On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091205.
Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.
Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).
As a composition of related permutations:
a(n) = A245703(A245822(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]
For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]

A091221 Number of distinct irreducible polynomials dividing n-th GF(2)[X]-polynomial.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 3, 2

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials

a(n) = A001221(A091203(n)) = A001221(A091205(n)). A000374(n) = a(A000051(n)).

f:= proc(n) local L,P,R,i;
  L:= convert(n,base,2);
  P:= add(L[i]*X^(i-1),i=1..nops(L));
  R:= Factors(P) mod 2;
  nops(R[2]);
end proc:
map(f, [$1.200]); # Robert Israel, Oct 11 2024

A091220 Number of divisors of the n-th GF(2)[X]-polynomial.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 2, 6, 2, 4, 4, 5, 5, 8, 2, 9, 3, 4, 4, 8, 2, 4, 6, 6, 4, 8, 2, 6, 4, 10, 4, 12, 2, 4, 6, 12, 2, 6, 4, 6, 8, 8, 2, 10, 4, 4, 6, 6, 4, 12, 2, 8, 6, 8, 2, 12, 2, 4, 6, 7, 9, 8, 2, 15, 3, 8, 4, 16, 2, 4, 8, 6, 4, 12, 4, 15, 3, 4, 8, 9, 7, 8, 2, 8, 4, 16, 2, 12, 4, 4, 6, 12, 2

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

a(n) = A000005(A091203(n)) = A000005(A091205(n)). Cf. A091257.

a(n)=local(p,fm,k);while(n>0,p+=Mod(n,2)*x^k;n\=2;k++);fm=factor(p);prod(k=1,matsize(fm)[1],fm[k,2]+1) \\ Franklin T. Adams-Watters, Jun 22 2010

A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 8, 5, 6, 16, 14, 17, 12, 19, 10, 28, 9, 11, 32, 13, 21, 20, 34, 53, 24, 49, 38, 64, 15, 43, 56, 59, 18, 68, 22, 35, 48, 37, 26, 76, 42, 67, 40, 29, 51, 112, 106, 107, 36, 25, 98, 44, 57, 23, 128, 119, 30, 52, 86, 31, 84, 131, 118, 80, 27, 133, 136, 41, 33

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2, and is self-inverse. It swaps 3 & 4, maps any prime p_i with index i > 2 to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 3 and 4, wherever they occur as the terminal configurations anywhere in the tree:
....................
.o..................
.|..................
.o.............o...o
.|..............\./.
.x.....<--->.....x..
.3...............4..
That is, the last two edges of any branch which ends with at least in two edges long unbranched stem, will be changed to a V-branch (two single edges in parallel). Vice versa, any terminal configuration in the tree that consists of more than one single edges next to each other (in "parallel") will be transformed so that maximal even number (2k) of those single edges will be combined to k unbranching stems of two edges, and an extra odd edge, if present, will stay as it is.
This permutation commutes with A235199, i.e. a(A235199(n)) = A235199(a(n)) for all n. This can be easily seen, when comparing the above bijection to the one described in A235199. Composition A235199 o A235201 works as a "difference" of these two bijections, swapping the above subconfigurations only when they do not occur alone at the tips of singular edges. (Which cases are encoded by Matula-Goebel numbers 5 and 7, the third and fourth prime respectively).
Permutation fixes n! for n=0, 1, 2, 4, 7.
Note that a(5!) = a(120) = 168 = 120+(2*4!) and a(8!) = a(40320) = 30240 = 40320-(2*7!).

Composition with A235487 gives A235485/A235486, composition with A235489 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A072026, A061773.

Multiplicative with a(3^k) = 2^(2k), a(2^(2k)) = 3^k, a(2^(2k+1)) = 2*3^k, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 17, 12, 13, 10, 21, 16, 11, 18, 19, 28, 15, 34, 23, 24, 49, 26, 27, 20, 43, 42, 59, 32, 51, 22, 35, 36, 37, 38, 39, 56, 41, 30, 29, 68, 63, 46, 73, 48, 25, 98, 33, 52, 53, 54, 119, 40, 57, 86, 31, 84, 61, 118, 45, 64, 91, 102

Offset: 0

Antti Karttunen, Jan 04 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=3 or 4, and is self-inverse. It swaps 5 & 7, maps all larger primes p_i (with index i > 4) to p_{a(i)}, and lets the multiplicativity take care of the rest.
It can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). The bijection will swap the subtrees encoded by primes 5 and 7, wherever they occur as the terminal branches of the tree:
....................
.o..................
.|..................
.o.............o...o
.|..............\./.
.o.....<--->.....o..
.|...............|..
.x...............x..
.5...............7..
That is, any branch which ends at least in three edges long unbranched stem, will be changed so that its last two edges will become V-branch. Vice versa, any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will be straightened to an unbranching stem of three edges.
This permutation commutes with A235201, i.e. a(A235201(n)) = A235201(a(n)) for all n.
Permutation fixes n! for n=0, 1, 2, 3, 4, 7, 8 and 9.
Note also that a(5!) = a(120) = 168 = 120+(2*4!) and a(10!) = 5080320 = 3628800+(4*9!).

Composition with A234840 gives A234743 & A234744.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A235485/A235486, A235493/A235494, A091204/A091205, A072026, A061773.

For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

A091219 Moebius-analog for the domain GF(2)[X]: a(n)=0 if A091221(n)!=A091222(n) (i.e., if the polynomial is not squarefree), otherwise (-1)^A091222(n).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0, -1, 1, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, -1, 0, 0, -1, -1, 0, -1, 1, 0, 0, 0, -1, -1, 0, 0, -1, 1, 0, -1, 1, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, -1, -1, 0, 1, 0

Offset: 1

Antti Karttunen, Jan 03 2004

sign

The absolute values give a characteristic function for squarefree GF(2)[X]-polynomials.

a(n) = A008683(A091203(n)) = A008683(A091205(n)).

A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 18, 17, 16, 23, 20, 21, 22, 19, 27, 25, 26, 24, 28, 29, 30, 31, 36, 33, 34, 35, 32, 37, 46, 39, 45, 41, 42, 43, 44, 40, 38, 47, 54, 49, 50, 51, 52, 61, 48, 55, 63, 69, 58, 59, 60, 53, 62, 56, 81, 65, 66, 83, 68, 57, 70, 71, 72, 73, 74, 75, 92, 77, 78, 79, 90, 64

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n, and is self-inverse. It swaps 8 & 9, maps any prime p_i with index i to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 8 and 9, wherever they occur as the terminal branches of the tree:
.......................
.................o...o.
.................|...|.
.o.o.o...........o...o.
..\|/.............\./..
...x.....<--->.....x...
...8...............9...
Thus, any terminal configuration in the tree that consists of three or more single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be replaced by k subtrees Matula-Goebel-encoded by 9 (see above, or equally: replaced by 2k two-edges-long branches encoded by 3), and one or two left-over single edges, if present, will stay as they are. Vice versa, any terminal configuration in the tree that consists of more than one two-edges-long branches next to each other (in "parallel") will be transformed so that maximal even number (2k) of those double-edges will be replaced by 3k single edges, and an extra odd double-edge, if present, will stay as it is.
Note how in contrast to A235487, A235201 and A235199, this bijection is not size-preserving (the number of edges will change), which has implications when composing this with other such permutations (cf. e.g. A235493/A235494).

Composition with A235201 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(3^(2k)) = 2^3k = 8^k, a(3^(2k+1)) = 3*2^3k, a(2^(3k)) = 3^2k = 9^k, a(2^(3k+1)) = 2*9^k, a(2^(3k+2)) = 4*9^k, a(p_i) = p_{a(i)} for primes with index i, and a(u*v) = a(u) * a(v) for composites other than 8 or 9.

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 16, 15, 14, 19, 18, 17, 20, 24, 22, 23, 21, 25, 26, 27, 32, 29, 30, 31, 28, 33, 38, 40, 36, 37, 34, 39, 35, 41, 48, 53, 44, 45, 46, 47, 42, 64, 50, 57, 52, 43, 54, 55, 56, 51, 58, 67, 60, 61, 62, 72, 49, 65, 66, 59, 76, 69, 80, 71, 63, 89, 74, 75, 68, 88, 78, 79, 70, 81

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=4, and is self-inverse. It swaps 7 & 8, maps any prime p_i with index i > 4 to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 7 and 8, wherever they occur as the terminal configurations anywhere in the tree:
.......................
.o...o.................
..\./..................
...o.............o.o.o.
...|..............\|/..
...x.....<--->.....x...
...7...............8...
Thus any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will emanate "in parallel" from the same vertex. Vice versa, any terminal configuration in the tree that consists of more than two single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be transformed to k Y-formations, and one or two left-over edges, if present, will stay as they are.

Composition with A235201 gives A235485/A235486.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(p_i) = p_{a(i)} for primes with index i <> 4, a(7) = 8, a(2^(3k)) = 7^k, a(2^(3k+1)) = 2*7^k, a(2^(3k+2)) = 4*7^k, and for other composites, a(u * v) = a(u) * a(v).

A091230 Iterates of A014580, starting with a(0) = 1, a(n) = A014580^(n)(1). [Here A014580^(n) means the n-th fold application of A014580].

Original entry on oeis.org

1, 2, 3, 7, 25, 137, 1123, 13103, 204045, 4050293, 99440273

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Cf. A000079, A007097, A091238, A091204-A091205, A245701-A245702, A245703-A245704.

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
prev=1; i=0; print1(1, ", "); for(n=1, 123456789, if(isA014580(n), i++; if((i == prev), print1(n, ", "); prev=n))) \\ Antti Karttunen, Aug 02 2014

a(0)=1, a(n) = A014580(a(n-1)). [The defining recurrence].
From Antti Karttunen, Aug 03 2014: (Start)
Other identities. For all n >= 0, the following holds:
A091238(a(n)) = n+1.
a(n) = A091204(A007097(n)) and A091205(a(n)) = A007097(n).
a(n) = A245703(A007097(n)) and A245704(a(n)) = A007097(n).
a(n) = A245702(A000079(n)) and A245701(a(n)) = A000079(n).
(End)

Terms a(8)-a(10) computed by Antti Karttunen, Aug 02 2014

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A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

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A091221 Number of distinct irreducible polynomials dividing n-th GF(2)[X]-polynomial.

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A091220 Number of divisors of the n-th GF(2)[X]-polynomial.

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A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

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A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

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A091219 Moebius-analog for the domain GF(2)[X]: a(n)=0 if A091221(n)!=A091222(n) (i.e., if the polynomial is not squarefree), otherwise (-1)^A091222(n).

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(u*v) = a(u) * a(v).

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A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(u*v) = a(u) * a(v).

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A091230 Iterates of A014580, starting with a(0) = 1, a(n) = A014580^(n)(1). [Here A014580^(n) means the n-th fold application of A014580].

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).