A091203 -id:A091203 - cp's OEIS frontend

A278233 Filter-sequence for GF(2)[X]-factorization: sequence that gives the least natural number with the same prime signature that (0, 1)-polynomial encoded in the binary expansion of n has when it is factored over GF(2).

1, 2, 2, 4, 4, 6, 2, 8, 6, 12, 2, 12, 2, 6, 8, 16, 16, 30, 2, 36, 4, 6, 6, 24, 2, 6, 12, 12, 6, 24, 2, 32, 6, 48, 6, 60, 2, 6, 12, 72, 2, 12, 6, 12, 24, 30, 2, 48, 6, 6, 32, 12, 6, 60, 2, 24, 12, 30, 2, 72, 2, 6, 12, 64, 36, 30, 2, 144, 4, 30, 6, 120, 2, 6, 24, 12, 6, 60, 6, 144, 4, 6, 30, 36, 64, 30, 2, 24, 6, 120, 2, 60, 6, 6, 12, 96, 2, 30, 12, 12, 30, 96, 2

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

a(n) = the least number with the same prime signature as A091203(n).
This sequence works as an A046523-analog in the polynomial ring GF(2)[X] and can be used as a filter which matches with (and thus detects) any sequence in the database where a(n) depends only on the exponents of irreducible factors when the polynomial corresponding to n (via base-2 encoding) is factored over GF(2). These sequences are listed in the Crossrefs section, "Sequences that partition N into ...".
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

			3 is "11" in binary, encodes polynomial x + 1, and 7 is "111" in binary, encodes polynomial x^2 + x + 1, both which are irreducible over GF(2). We can multiply their codes with carryless multiplication A048720 as A048720(3,7) = 9, A048720(9,3) = 27, A048720(9,7) = 63. Now a(27) = a(63) because the exponents occurring in both codes 27 and 63 are one 1 and two 2's, and their order is not significant when computing prime signature. Moreover a(27) = a(63) = 12 because that is the least number with a prime signature (1,2) in the more familiar domain of natural numbers.
a(25) = 2, because 25 is "11001" in binary, encoding polynomial x^4 + x^3 + 1, which is irreducible in the ring GF(2)[X], i.e., 25 is in A014580, whose initial term is 2.

Cf. A014580 (gives the positions of 2's), A048720, A057889, A091203, A091205, A193231, A235042, A278231, A278238, A278239.
Similar filtering sequences: A046523, A278222, A278226, A278236, A278243.
Sequences that partition N into same or coarser equivalence classes: A091220, A091221, A091222, A106493, A106494.
Cf. also A304529, A304751, A305788 (rgs-transform), A305789.

A278233(n) = { my(p=0, f=vecsort((factor(Pol(binary(n))*Mod(1, 2))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ Antti Karttunen, Jun 10 2018

(define (A278233 n) (A046523 (A091203 n)))

a(n) = A046523(A091203(n)) = A046523(A091205(n)) = A046523(A235042(n)). [Because of the "sorting" essentially performed by A046523, any map from GF(2)[X] to Z can be used, as long as it is fully (cross-)multiplicative and preserves also the exponents intact.]
Other identities. For all n >= 1:
a(A014580(n)) = 2.
a(n) = a(A057889(n)) = a(A193231(n)).
a(A000695(n)) = A278238(n).
a(A277699(n)) = A278239(n).

A235042 Factorization-preserving bijection from GF(2)[X]-polynomials to nonnegative integers, version which fixes the elements that are irreducible in both semirings.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 7, 8, 21, 18, 11, 12, 13, 14, 27, 16, 81, 42, 19, 36, 49, 22, 39, 24, 5, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 15, 44, 189, 78, 47, 48, 77, 10, 243, 52, 57, 126, 17, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Like A091203 this is a factorization-preserving isomorphism from GF(2)[X]-polynomials to integers. The former are encoded in the binary representation of n like this: n=11, '1011' in binary, stands for polynomial x^3+x+1, n=25, '11001' in binary, stands for polynomial x^4+x^3+1. However, this version does not map the irreducible GF(2)[X] polynomials (A014580) straight to the primes (A000040), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A014580 \ A000040 (= A091214) in numerical order to the set-wise difference A000040 \ A014580 (= A091209).
The composite values are defined by the multiplicativity. E.g., we have a(A048724(n)) = 3*a(n) and a(A001317(n)) = A000244(n) = 3^n for all n.
This map satisfies many of the same identities as A091203, e.g., we have A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)) and A091247(n) = A066247(a(n)) for all n >= 1.

			Here (t X u) = A048720(t,u):
a(2)=2, a(3)=3 and a(7)=7, as 2, 3 and 7 are all in A091206.
a(4) = a(2 X 2) = a(2)*a(2) = 2*2 = 4.
a(5) = a(3 X 3) = a(3)*a(3) = 3*3 = 9.
a(9) = a(3 X 7) = a(3)*a(7) = 3*7 = 21.
a(10) = a(2 X 3 X 3) = a(2)*a(3)*a(3) = 2*3*3 = 18.
a(15) = a(3 X 3 X 3) = a(3)^3 = 3^3 = 27.
a(17) = a(3 X 3 X 3 X 3) = a(3)^4 = 3^4 = 81.
a(21) = a(7 X 7) = a(7)*a(7) = 7*7 = 49.
a(25) = 5, as 25 is the first term of A091214 and 5 is the first term of A091209.
a(50) = a(2 X 25) = a(2)*a(25) = 2*5 = 10.

Inverse: A235041. Fixed points: A235045.
Cf. A010051, A004247, A091225, A091209, A235044.
Similar cross-multiplicative permutations: A091203, A091205, A106443, A106445, A106447.

a(0)=0, a(1)=1, a(p) = p for those irreducible GF(2)[X]-polynomials whose binary encoding is a prime (i.e., p is in A091206), and for the rest of irreducible GF(2)[X]-polynomials (those which are encoded by a composite natural number, i.e., q is in A091214), a(q) = A091209(A235044(q)), and for reducible polynomials, a(i X j X k X ...) = a(i) * a(j) * a(k) * ..., where each i, j, k, ... is in A014580, X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).

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].

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

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

Antti Karttunen, Jan 03 2004

nonn

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

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

A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 768, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47

Offset: 0

Antti Karttunen, May 09 2005

nonn

This map from the multiplicative domain of GF(2)[X] to that of N preserves Catalan-family structures, e.g. A075164(n) = a(A106454(n)), A106453(n) = A075163(a(n)), A106455(n) = A075165(a(n)), A106456(n) = A075166(a(n)), A106457(n) = A075167(a(n)). Shares with A091203 and A106445 the property that maps A014580(n) to A000040(n). Differs from the former for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the latter for the first time at n=48, where A106445(48)=48, while a(48)=768.

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^(a(4+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.

Inverse: A106442. a(n) = A075164(A106453(n)).

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^(a(1+e_j)-1) * a(ir_k)^(a(1+e_k)-1) * ... = A000040(i)^a(e_i) * A000040(j)^(a(1+e_j)-1) * A000040(k)^(a(1+e_k)-1), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation. Here ir_i is the most significant (largest) irreducible polynomial in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant factors e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

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)).

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

Antti Karttunen, Jun 10 2018

nonn

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

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

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))));

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).

A305903 Filter sequence for all such sequences b, for which b(A014580(k)) = constant for all k >= 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 11, 7, 12, 13, 14, 15, 16, 7, 17, 18, 19, 20, 21, 7, 22, 23, 24, 25, 26, 7, 27, 28, 29, 30, 31, 7, 32, 33, 34, 7, 35, 36, 37, 38, 39, 7, 40, 41, 42, 43, 44, 45, 46, 7, 47, 48, 49, 7, 50, 7, 51, 52, 53, 54, 55, 7, 56, 57, 58, 59, 60, 7, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 7, 74, 75, 76, 7, 77, 78, 79, 80, 81, 7

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A305900(A091203(n)).
This is GF(2)[X] analog of A305900.
For all i, j:
  a(i) = a(j) => A304529(i) = A304529(j) => A305788(i) = A305788(j).
  a(i) = a(j) => A268389(i) = A268389(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A091203, A268389, A278233, A304529, A305788, A305900.

up_to = 1000;
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
prepare_v091226(up_to) = { my(v = vector(up_to), c=0); for(i=1,up_to,c += A091225(i); v[i] = c); (v); }
v091226 = prepare_v091226(up_to);
A091226(n) = if(!n,n,v091226[n]);
A305903(n) = if(n<7,n,if(A091225(n),7,3+n-A091226(n)));

For n < 7, a(n) = n, for >= 7, a(n) = 7 when n is in A014580[3..] (= 7, 11, 13, 19, 25, 31, ...), and a(n) = 3+n-A091226(n) when n is in A091242[4..] (= 8, 9, 10, 12, 14, 15, ...).

cp's OEIS Frontend

A278233 Filter-sequence for GF(2)[X]-factorization: sequence that gives the least natural number with the same prime signature that (0, 1)-polynomial encoded in the binary expansion of n has when it is factored over GF(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

A235042 Factorization-preserving bijection from GF(2)[X]-polynomials to nonnegative integers, version which fixes the elements that are irreducible in both semirings.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Links

Crossrefs

Formula

A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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