A091221 -id:A091221 - cp's OEIS frontend

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).

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)).

A001221 Number of distinct primes dividing n (also called omega(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A091202 Factorization-preserving isomorphism from nonnegative integers to binary codes for polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 13, 12, 19, 22, 9, 16, 25, 10, 31, 28, 29, 26, 37, 24, 21, 38, 15, 44, 41, 18, 47, 32, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 48, 69, 42, 43, 76, 73, 30, 35, 88, 33, 82, 87, 36, 91, 94, 39, 64, 121, 46, 97, 100, 111, 98

Offset: 0

Antti Karttunen, Jan 03 2004

E.g. we have the following identities: A000005(n) = A091220(a(n)), A001221(n) = A091221(a(n)), A001222(n) = A091222(a(n)), A008683(n) = A091219(a(n)), A014580(n) = a(A000040(n)), A049084(n) = A091227(a(n)).

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: A091203.
Several variants exist: A235041, A091204, A106442, A106444, A106446.
Cf. also A000005, A091220, A001221, A091221, A001222, A091222, A008683, A091219, A000040, A014580, A048720, A049084, A091227, A245703, A234741.
Cf. also A302023, A302025, A305417, A305427 for other similar permutations.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091202(n) = if(n<=1,n,if(!(n%2),2*A091202(n/2),A305421(A091202(A064989(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i 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).
Other identities. For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242. The permutations A091204, A106442, A106444, A106446, A235041 and A245703 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A305421(a(A064989(n))).
a(n) = A305417(A156552(n)) = A305427(A243071(n)).
(End)

A091203 Factorization-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, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

E.g. we have the following identities: A000040(n) = a(A014580(n)), 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)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)).

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: A091202.
Several variants exist: A235042, A091205, A106443, A106445, A106447.
Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233.
Cf. also A302024, A302026, A305418, A305428 for other similar permutations.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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); };
A091203(n) = if(n<=1,n,if(!(n%2),2*A091203(n/2),A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018

a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).
Other identities. 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, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.]
From Antti Karttunen, Jun 10 2018: (Start)
For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))).
a(n) = A005940(1+A305418(n)) = A163511(A305428(n)).
A046523(a(n)) = A278233(n).
(End)

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).

Original entry on oeis.org

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).

A235041 Factorization-preserving bijection from nonnegative integers to GF(2)[X]-polynomials, version which fixes the elements that are irreducible in both semirings.

Original entry on oeis.org

0, 1, 2, 3, 4, 25, 6, 7, 8, 5, 50, 11, 12, 13, 14, 43, 16, 55, 10, 19, 100, 9, 22, 87, 24, 321, 26, 15, 28, 91, 86, 31, 32, 29, 110, 79, 20, 37, 38, 23, 200, 41, 18, 115, 44, 125, 174, 47, 48, 21, 642, 89, 52, 117, 30, 227, 56, 53, 182, 59, 172, 61, 62, 27, 64

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Like A091202 this is a factorization-preserving isomorphism from integers to GF(2)[X]-polynomials. The latter 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 primes (A000040) straight to the irreducible GF(2)[X] polynomials (A014580), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A000040 \ A014580 (= A091209) in numerical order to the set-wise difference A014580 \ A000040 (= A091214).
The composite values are defined by the multiplicativity. E.g., we have a(3n) = A048724(a(n)) and a(3^n) = A001317(n) for all n.
This map satisfies many of the same identities as A091202, e.g., we have A000005(n) = A091220(a(n)), A001221(n) = A091221(a(n)), A001222(n) = A091222(a(n)) and A008683(n) = A091219(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*2) = a(2) X a(2) = 2 X 2 = 4.
a(9) = a(3*3) = a(3) X a(3) = 3 X 3 = 5.
a(5) = 25, as 5 is the first term of A091209 and 25 is the first term of A091214.
a(10) = a(2*5) = a(2) X a(5) = 2 X 25 = 50.
Similarly, a(17) = 55, as 17 is the second term of A091209 and 55 is the second term of A091214.
a(21) = a(3*7) = a(3) X a(7) = 3 X 7 = 9.

Inverse: A235042. Fixed points: A235045.
Cf. A010051, A048720, A091225, A091214, A235043, A048724, A115857, A115872.
Similar cross-multiplicative permutations: A091202, A091204, A106442, A106444, A106446.

a(0)=0, a(1)=1, a(p) = p for those primes p whose binary representations encode also irreducible GF(2)[X]-polynomials (i.e., p is in A091206), and for the rest of the primes q (those whose binary representation encode composite GF(2)[X]-polynomials, i.e., q is in A091209), a(q) = A091214(A235043(q)), and for composite natural numbers, a(p * q * r * ...) = a(p) X a(q) X a(r) X ..., where p, q, r, ... are primes and X stands for the carryless multiplication (A048720) of GF(2)[X] polynomials encoded as explained in the Comments section.

A106495 Number of leaves in GF(2)[X] superfactorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 09 2005

nonn

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A064372(A106445(n)). After n=1 differs from A091221 for the first time at n=64, where A091221(64)=1, while a(64)=2.

a(n) = A106494(n)-A106493(n).

A304107 Analog for squarefree numbers when n is factored in polynomial ring GF(2)[X], so that the binary expansion of n defines the corresponding (0,1)-polynomial. These are numbers n such that the said polynomial doesn't have any duplicated irreducible divisors.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 11, 13, 14, 18, 19, 22, 23, 25, 26, 29, 31, 33, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 70, 71, 73, 74, 77, 79, 82, 83, 86, 87, 89, 91, 93, 94, 97, 98, 101, 103, 106, 109, 110, 111, 113, 115, 117, 118, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 142, 143, 145, 146, 149, 154, 155, 157, 158, 159, 161

Offset: 1

Antti Karttunen, May 13 2018

nonn

Positions of nonzeros in A091219 and A304109. Numbers n such that A091221(n) = A091222(n).
Numbers n that cannot be expressed as n = A048720(k,A000695(m)) for any k >= 0, m >= 2.
It seems that a(n) is approximately 2n for large n. See also comments in A304110.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A304108 (complement), A304109 (characteristic function), A304110 (least monotonic left inverse).
Cf. A000695, A014580, A048720, A091219, A091221, A091222, A304111.
Cf. also A005117.

A304109(n) = { my(fm=factor(Pol(binary(n))*Mod(1, 2))); for(k=1, #fm~, if(fm[k, 2] > 1, return(0))); (1); };
k=0; n=0; while(k<100, n++; if(A304109(n), k++; print1(n,", ")));

For n >= 1, A304110(a(n)) = 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).

cp's OEIS Frontend

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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A001221 Number of distinct primes dividing n (also called omega(n)).

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A091202 Factorization-preserving isomorphism from nonnegative integers to binary codes for polynomials over GF(2).

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A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

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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).

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

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A235041 Factorization-preserving bijection from nonnegative integers to GF(2)[X]-polynomials, version which fixes the elements that are irreducible in both semirings.

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