A091245 -id:A091245 - cp's OEIS frontend

A245814 Permutation of natural numbers induced when A091204 is restricted to nonprime numbers: a(n) = 1+A091245(A091204(A018252(n))).

1, 2, 4, 5, 3, 9, 8, 16, 6, 11, 7, 21, 22, 39, 18, 15, 29, 10, 34, 13, 24, 33, 76, 38, 14, 48, 42, 44, 46, 81, 20, 19, 37, 54, 32, 92, 60, 23, 63, 71, 25, 99, 28, 233, 30, 50, 98, 70, 157, 17, 79, 31, 89, 49, 101, 191, 86, 91, 12, 161, 94, 171, 193, 56, 167, 43, 143, 41, 353, 58, 75, 78, 113, 102, 68, 190, 125, 67, 119, 47, 130, 72, 146, 52, 27

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

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

Inverse: A245813.
Related permutations: A091204, A245816, A245819.
Cf. A018252, A091245.

allocatemem(123456789);
v014580 = vector(2^18);
v091226 = 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++; v091226[n] = v091226[n-1]); n++);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler
A018252(n) = if(1==n, 1, A002808(n-1));
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091245(n) = ((n-A091226(n))-1);
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))}));
A245814(n) = 1 + A091245(A091204(A018252(n)))
for(n=1, 10001, write("b245814.txt", n, " ", A245814(n)));

(define (A245814 n) (+ 1 (A091245 (A091204 (A018252 n)))))

a(n) = 1 + A091245(A091204(A018252(n))).
As a composition of related permutations:
a(n) = A245819(A245816(n)).

A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 12, 7, 9, 13, 26, 10, 14, 18, 11, 15, 48, 19, 20, 35, 16, 21, 32, 25, 17, 22, 63, 27, 56, 28, 138, 46, 23, 29, 43, 34, 38, 24, 30, 80, 60, 36, 88, 72, 37, 167, 42, 59, 31, 39, 55, 45, 62, 50, 33, 40, 100, 77, 320, 47, 92, 109, 90, 49, 201, 54, 98, 76, 41, 51

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

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

Inverse: A245820.
Related permutations: A245703, A245814, A245815.
Cf. A008578, A018252, A091226, A091245.

\\ The rest of code can be found in A245703.
A245819(n) = if(1==n, 1, 1 + A245703(n-1));

(define (A245819 n) (if (<= n 1) n (+ 1 (A245703 (- n 1)))))

(define (A245819 n) (+ 1 (A091245 (A245703 (A018252 n)))))

(define (A245819 n) (+ 1 (A091226 (A245703 (A008578 n)))))

a(1) = 1, and for n > 1, a(n) = 1 + A245703(n-1).
a(n) = 1+A091245(A245703(A018252(n))). [Induced when A245703 is restricted to nonprime numbers].
a(n) = 1+A091226(A245703(A008578(n))). [Induced also when A245703 is restricted to noncomposite numbers].
As a composition of related permutations:
a(n) = A245814(A245815(n)).

A091247 Characteristic function of A091242: 1 if the n-th GF(2)[X] polynomial is reducible, 0 otherwise.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
A. Karttunen, Scheme-program for computing this sequence.
Index entries for characteristic functions
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A066247, A091203, A091205, A091246.

Complementary to A091225. Partial sums give A091245.

a(n) = if (n<2, 0, ! polisirreducible(Mod(1,2)*Pol(binary(n)))); \\ Michel Marcus, Apr 12 2015

a(n) = A066247(A091203(n)) = A066247(A091205(n)).

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

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].

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(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

Antti Karttunen, Aug 02 2014

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

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

allocatemem(123456789);
A091226 = 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), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245701(n) = if(1==n, 1, if(isA014580(n), 2*(A245701(A091226[n])), 1 + 2*(A245701(A091245(n)))));
for(n=1, 10001, write("b245701.txt", n, " ", A245701(n)));

;; With memoizing definec-macro.
(definec (A245701 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (* 2 (A245701 (A091226 n)))) (else (+ 1 (* 2 (A245701 (A091245 n)))))))

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].

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

Antti Karttunen, Aug 02 2014

nonn

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

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

allocatemem(123456789);
default(primelimit, 2^22)
v014580 = vector(2^18); A014580(n) = v014580[n];
v091226 = vector(2^22); A091226(n) = v091226[n];
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
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); 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))}));
A091245(n) = ((n-A091226(n))-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226(n))), A002808(A245704(A091245(n)))));
A245822(n) = A245704(A091204(n));
for(n=1, 10001, write("b245822.txt", n, " ", A245822(n)));

(define (A245822 n) (A245704 (A091204 n)))

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)) = (2a(n))+1, a(A091242(n)) = 2a(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

Antti Karttunen, Aug 19 2014

nonn

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).

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

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].

A246163 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A065621(1+a(n)), a(A091242(n)) = A048724(a(n)), where A065621(n) and A048724(n) give the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 3, 6, 9, 8, 5, 10, 27, 4, 24, 11, 15, 30, 45, 12, 40, 26, 29, 17, 34, 119, 20, 25, 120, 46, 39, 51, 102, 14, 153, 60, 43, 136, 114, 31, 105, 85, 170, 44, 18, 427, 68, 125, 408, 13, 150, 33, 187, 255, 510, 116, 54, 41, 765, 204, 135, 28, 680, 16, 23, 442, 99, 461, 257, 35, 514, 156, 90, 123, 1799, 118, 340, 393, 36

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)) and A065621/A048724, the latter which themselves are permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246161.
Because 3 is the only evil number in A014580, it implies that, apart from a(3)=7, odious numbers occur in odious positions only (along with many evil numbers that also occur in odious positions).
Furthermore, all terms of A246158 reside in infinite cycles, and apart from 4 and 8, all of them reside in separate cycles. The infinite cycle containing 4 and 8 goes as: ..., 2091, 97, 47, 13, 11, 4, 3, 7, 8, 5, 6, 9, 10, 27, 46, 408, 2535, ... and it is only because a(3) = 7, that it can temporarily switch back from evil terms to odious terms, until right after a(8) = 5 it is finally doomed to the eternal evilness.
Please see also the comments at A246201 and A246161.

Inverse: A246164.
Similar or related permutations: A246205, A193231, A246201, A234026, A245701, A234612, A246161, A246203.
Cf. A010060, A014580, A048724, A065621, A091225, A091226, A091245.

a(1) = 1, and for n > 1, if n is in A014580, a(n) = A065621(1+a(A091226(n))), otherwise a(n) = A048724(a(A091245(n))).
As a composition of related permutations:
a(n) = A193231(A246201(n)).
a(n) = A234026(A245701(n)).
a(n) = A234612(A246161(n)).
a(n) = A193231(A246203(A193231(n))).
Other identities:
For all n > 1, A010060(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (A014580) to odious numbers and the corresponding representations of reducible polynomials (A091242) to evil numbers, in some order].

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).

A236863 Number of nonludic numbers (A192607) not greater than n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 07 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A065855, A091245, A237427.

(define (A236863 n) (if (zero? n) n (- n (A192512 n))))

a(0)=0, a(n) = n - A192512(n).

cp's OEIS Frontend

A245814 Permutation of natural numbers induced when A091204 is restricted to nonprime numbers: a(n) = 1+A091245(A091204(A018252(n))).

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A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

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A091247 Characteristic function of A091242: 1 if the n-th GF(2)[X] polynomial is reducible, 0 otherwise.

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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).

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

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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).

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

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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(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).

A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2a(n))+1, a(A091242(n)) = 2a(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).