A236850 -id:A236850 - cp's OEIS frontend

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

2, 3, 7, 11, 13, 19, 25, 31, 37, 41, 47, 55, 59, 61, 67, 73, 87, 91, 97, 103, 109, 115, 117, 131, 137, 143, 145, 157, 167, 171, 185, 191, 193, 203, 211, 213, 229, 239, 241, 247, 253, 283, 285, 299, 301, 313, 319, 333, 351, 355, 357, 361, 369, 375

Offset: 1

David Petry (petry(AT)accessone.com)

nonn

Or, binary irreducible polynomials, interpreted as binary vectors, then written in base 10.
The numbers {a(n)} are a subset of the set {A206074}. - Thomas Ordowski, Feb 21 2014
2^n - 1 is a term if and only if n = 2 or n is a prime and 2 is a primitive root modulo n. - Jianing Song, May 10 2021
For odd k, k is a term if and only if binary_reverse(k) = A145341((k+1)/2) is. - Joerg Arndt and Jianing Song, May 10 2021

			x^4 + x^3 + 1 -> 16+8+1 = 25. Or, x^4 + x^3 + 1 -> 11001 (binary) = 25 (decimal).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (first 1377 terms from T. D. Noe)
Index entries for sequences operating on GF(2)[X]-polynomials

Written in binary: A058943.
Number of degree-n irreducible polynomials: A001037, see also A000031.
Multiplication table: A048720.
Characteristic function: A091225. Inverse: A091227. a(n) = A091202(A000040(n)). Almost complement of A091242. Union of A091206 & A091214 and also of A091250 & A091252. First differences: A091223. Apart from a(1) and a(2), a subsequence of A092246 and hence A000069.
Table of irreducible factors of n: A256170.
Irreducible polynomials satisfying particular conditions: A071642, A132447, A132449, A132453, A162570.
Factorization sentinel: A278239.
Sequences analyzing the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the corresponding integer: A234741, A234742, A235032, A235033, A235034, A235035, A235040, A236850, A325386, A325559, A325560, A325563, A325641, A325642, A325643.
Factorization-preserving isomorphisms: A091203, A091204, A235041, A235042.
See A115871 for sequences related to cross-domain congruences.
Functions based on the irreducibles: A305421, A305422.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n]; fQ[2] = True; Select[ Range@ 378, fQ] (* Robert G. Wilson v, Aug 12 2011 *)
Reap[Do[If[IrreduciblePolynomialQ[IntegerDigits[n, 2] . x^Reverse[Range[0, Floor[Log[2, n]]]], Modulus -> 2], Sow[n]], {n, 2, 1000}]][[2, 1]] (* Jean-François Alcover, Nov 21 2016 *)

is(n)=polisirreducible(Pol(binary(n))*Mod(1,2)) \\ Charles R Greathouse IV, Mar 22 2013

A234742 Product of the binary encodings of the irreducible factors (with multiplicity) of the polynomial over GF(2) whose encoding is n.

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, 25, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 75, 44, 189, 78, 47, 48, 77, 50, 243, 52, 57, 126, 55, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441

Offset: 0

Antti Karttunen, Jan 22 2014

"Product" refers to the ordinary multiplication of integers.
Differs from A235042 and A236837 for the first time at n=25, where a(n)=25, while A235042(25)=5 and A236837(25)=0. Thus A234741(A234742(n)) = n up to n=24.
a(n) >= n. [All terms of the table A061858 are nonnegative as the product of multiplying two numbers with carries is never less than when multiplying them without carries.]
Specifically, for all n, a(A091209(n)) > A091209(n).
a(A091209(n)) is always composite and, by the above inequality, larger than A091209(n), which implies that none of the terms of A091209 occur in this sequence. Cf. also A236844.
Starting with various terms (primes) in A235033 and iterating the map A234742, we get 5 -> 9 -> 21 -> 49 -> 77 -> 177 -> 333 = a(333).
Another example: 17 -> 81 -> 169 -> 309 -> 721 = a(721).
Does every chain of such iterations eventually reach a fixed point? (One of the terms of A235035.) Or do some of them manage to avoid such "traps" indefinitely? (Note how the terms of A235035 seem to get rarer, but only rather slowly.)
Starting from 23, we get the sequence: 23, 39, 99, 279, 775, 1271, 3003, 26411, 45059, ... which reaches its fixed point, 3643749709604450870616156947649219, after 55 iterations. - M. F. Hasler, Feb 18 2014. [This is now sequence A244323. See also A260729, A260735 and A260441.] - Antti Karttunen, Aug 05 2015
Note also that when coming backwards from some term of such a chain by iterating A234741, we may not necessarily end at the same term we started from.

			3 has binary representation '11', which encodes the polynomial X + 1, which is irreducible in GF(2)[X], so the result is just a(3)=3.
5 has binary representation '101' which encodes the polynomial X^2 + 1, which is reducible in the polynomial ring GF(2)[X], factoring as (X+1)(X+1), i.e., 5 = A048720(3,3), as 3 ('11' in binary) encodes the polynomial (X+1), irreducible in GF(2)[X]. 3*3 = 9, thus a(5)=9.
9 has binary representation '1001', which encodes the polynomial X^3 + 1, which factors (in GF(2)[X]!) as (X+1)(X^2+X+1), i.e., 9 = A048720(3,7) (7, '111' in binary, encodes the other factor polynomial X^2+X+1). 3*7 = 21, thus a(9)=21.
25 has binary representation '11001', which encodes the polynomial X^4 + X^3 + 1, which is irreducible in GF(2)[X], so the result is just a(25)=25.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

A235035 gives the k for which a(k)=k.
A236853(n) gives the number of times n occurs in this sequence.
A236842 gives the same sequence sorted and with duplicates removed, A236844 gives the numbers that do not occur here, A236845 gives numbers that occur more than once, A236846 the least inverse and A236847 the greatest inverse. A236850 gives such k that a(k) = A236837(k).
Cf. A014580, A048720, A234741, A236379, A236837, A236851 & A236852, A115857, A115872.
Cf. also A260712, A260713, A260716 and A244323, A260729, A260735, A260441 (iterations starting from various terms of A236844).

A234742(n)=factorback(subst(lift(factor(Mod(1,2)*Pol(binary(n)))),x,2)) \\ M. F. Hasler, Feb 18 2014, corrected Andrew Howroyd, Aug 01 2018

To compute a(n): factor the polynomial over GF(2) encoded by n, into its irreducible factors; in other words, find a unique multiset of terms i, j, ..., k (not necessarily distinct) from A014580 for which i x j x ... x k = n, where x stands for the carryless multiplication A048720. Then a(n) = i*j*...*k is the product of those terms with ordinary multiplication. Because of the effect of the carry-bits in the latter, the result is always greater than or equal to n, so we have a(n) >= n for all n.
a(2n) = 2*a(n).
a(A235035(n)) = A235035(n).
A236379(n) = a(n) - n.
For all n, a(n) >= A236837(n).

A091214 Composite numbers whose binary representation encodes a polynomial irreducible over GF(2).

Original entry on oeis.org

25, 55, 87, 91, 115, 117, 143, 145, 171, 185, 203, 213, 247, 253, 285, 299, 301, 319, 333, 351, 355, 357, 361, 369, 375, 391, 395, 415, 425, 445, 451, 471, 477, 501, 505, 515, 529, 535, 539, 545, 623, 637, 665, 675, 687, 695, 721, 731, 789, 799, 803, 817

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).

Intersection of A002808 and A014580.
Subsequence of A235033, A236834 and A236838.
Left inverse: A235044.
Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091209 (Primes that encode a polynomial reducible over GF(2)), A091212 (Composite, and reducible over GF(2)).
Cf. A091215, A235027, A235046, A236841, A236845, A236850, A236851, A236861.
Cf. also A235041-A235042.

fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n && ! PrimeQ@ n]; Select[ Range@ 840, fQ] (* Robert G. Wilson v, Aug 12 2011 *)

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091214(n) = (!isprime(n) && isA014580(n));
n = 0; i = 0; while(n < 2^20, n++; if(isA091214(n), i++; write("b091214.txt", i, " ", n)));
\\ The b-file was computed with this program. Antti Karttunen, May 17 2015

Other identities. For all n >= 1:
A235044(a(n)) = n. [A235044 works as a left inverse of this sequence.]
a(n) = A014580(A091215(n)). - Antti Karttunen, May 17 2015

Entry revised and name corrected by Antti Karttunen, May 17 2015

A236837 The greatest inverse of A234741: a(n) = the largest k such that A234741(k) = n, and 0 if no such k exists.

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, 0, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 43, 44, 189, 78, 47, 48, 77, 0, 243, 52, 57, 126, 0, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441, 186, 67, 324, 121

Offset: 0

Antti Karttunen, Jan 31 2014

nonn

A234741(a(n)) = n, unless n is in A236834, in which case a(n)=0.
For all n, a(n) <= A234742(n). A236850 gives such k that a(k) = A234742(k).
If n is in A236835, a(n) > A236836(n), otherwise a(n) = A236836(n).
a(2^n) = 2^n.
a(2n) = 2*a(n).

Antti Karttunen, Table of n, a(n) for n = 0..8192

A236834 gives the positions of zeros.
Differs from A235042 and A234742 for the first time at n=25, where a(25)=0 but A235042(25)=5 and A234742(25)=25.
Cf. A236836 (the least inverse of A234741).
Cf. also A236833, A236835, A234741, A236841, A236850.

A236851 Remultiply n first "upward", from GF(2)[X] to N, and then remultiply that result back "downward", from N to GF(2)[X]: a(n) = A234741(A234742(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 51, 44, 45, 46, 47, 48, 49, 34, 51, 52, 53, 54, 39, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Antti Karttunen, Feb 02 2014

nonn

			5 ('101' in binary) = 3 x 3 (3 = '11' in binary). 3 is in A091206, so it stays intact, and 3 x 3 = 5, thus a(5)=5.
25 ('11001' in binary) = 25 (25 is irreducible in GF(2)[X]). However, it divides as 5*5 in Z, so the result is 5 x 5 = 17, thus a(25)=17, 25 being the least n which is not fixed by this function.
43 ('101011' in binary) = 3 x 25, of which the latter factor divides to 5*5, thus the result is 3 x 5 x 5 = 3 x 17 = 15 x 5 = 51.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences operating on polynomials over GF(2)

A236850 gives the fixed points.

Cf. A234741, A234742, A236380, A236852, A236836-A236837, A236846-A236847.

(define (A236851 n) (A234741 (A234742 n)))

a(n) = A234741(A234742(n)).
To compute a(n): factor n as a polynomial over GF(2) (where n is mapped to such polynomials via the binary representation of n), that is, find first a unique multiset of terms i, j, ..., k from A014580 for which i x j x ... x k = n, where x stands for the carryless multiplication (A048720). Then divide from those i, j, ..., k the ones that are in A091214 (composite integers in N) to their constituent prime factors (in N), and multiply all back together (including the factors that are in A091206 and thus not changed) with the carryless multiplication (A048720).
Compare this to how primes are "broken" in a similar way in A235027 (cf. also A235145).

A236860 After 0 and 1, numbers n all of whose prime divisors encode an irreducible polynomial over GF(2) (are terms of A091206).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 21, 22, 24, 26, 27, 28, 31, 32, 33, 36, 37, 38, 39, 41, 42, 44, 47, 48, 49, 52, 54, 56, 57, 59, 61, 62, 63, 64, 66, 67, 72, 73, 74, 76, 77, 78, 81, 82, 84, 88, 91, 93, 94, 96, 97, 98, 99, 103, 104, 108, 109

Offset: 1

Antti Karttunen, Mar 08 2014

nonn

After 0 and 1, positive integers which are products of p * q * ... * r, where p, q, ..., r are terms of A091206.
Also fixed points of A236852(n). Proof: if k is a term of this sequence, the operation described in A236852 reduces to an identity operation. On the other hand, if k is not a term of this sequence, then it has at least one prime divisor which is reducible in polynomial ring GF(2)[X], which is thus "broken" by A236852 (A234742) to two or more separate factors (either prime or not), and because the original factor was prime, and N is a unique factorization domain, the new product computed over the new set of factors (with one or more "broken" pieces) cannot be equal to the original k. (Compare this to how primes are "broken" in a similar way in A235027, also A235145.)
Note: This sequence is not equal to all n for which A234741(n) = A236846(n). The first counterexample occurs at a(325) = 741 (= 3*13*19) for which we have: A236846(741) = 281 (= 3 x 247 = 3 x (13*19)) while A234741(741) = 329 (= 3 x 13 x 19). Contrast this with the behavior of the "dual sequence" A236850, where the corresponding property holds.

Antti Karttunen, Table of n, a(n) for n = 1..12696

Complement: A236848.
Subsequence of A236842.
Fixed points of A236852.
Cf. A091206, A236850.

isp(p) = polisirreducible( Mod(1, 2) * Pol(binary(p))); \\ A091206
isok(n) = if ((n==0), 1 , my(f=factor(n)); prod(k=1, #f~, isp(f[k,1])) != 0); \\ Michel Marcus, Dec 22 2018

A236838 Numbers whose binary representation encodes a polynomial over GF(2) with the property that at least one of its irreducible factors is encoded by a composite number.

Original entry on oeis.org

25, 43, 50, 55, 79, 86, 87, 89, 91, 100, 110, 115, 117, 125, 133, 135, 143, 145, 149, 158, 159, 171, 172, 174, 178, 181, 182, 185, 200, 203, 209, 213, 220, 227, 230, 234, 235, 237, 247, 249, 250, 253, 263, 266, 267, 270, 279, 281, 285, 286, 290, 293, 298, 299

Offset: 1

Antti Karttunen, Feb 02 2014

nonn

Numbers which are of the form A048720(a,A091214(b)) for some a, b.

In the range 1..10000 about half of the natural numbers seem to be in this set, and the terms are getting more frequent, although rather slowly. (Please see the graph.)

			25, in binary '11001', encodes polynomial x^4 + x^3 + 1, which is irreducible in polynomial ring GF(2)[X], but is composite in N, thus it is a term of this sequence.
43, in binary '101011', encodes polynomial x^5 + x^3 + x + 1, which factors as (x + 1)(x^4 + x^3 + 1), i.e., 43 = A048720(3,25), and the latter factor of these, encoded by 25, is a composite in N, thus 43 is a term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Disjoint union of A236834 and A236839.
Complement: A236850.
Cf. A236848, A048720, A091214.

(define A236838 (MATCHING-POS 1 1 (lambda (n) (any (lambda (p) (= 1 (A066247 p))) (GF2Xfactor n)))))

A236839 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number and have at least one irreducible polynomial factor encoded by a composite number.

Original entry on oeis.org

43, 79, 86, 89, 125, 135, 149, 158, 172, 178, 181, 209, 227, 235, 250, 263, 270, 281, 293, 298, 311, 316, 317, 331, 344, 349, 356, 362, 371, 383, 393, 399, 401, 418, 421, 443, 447, 454, 470, 479, 493, 500, 509, 519, 521, 523, 526, 540, 547, 562, 567, 577, 586, 596, 609

Offset: 1

Antti Karttunen, Feb 02 2014

nonn

Terms of A236841 that are found in A236838.

			43, which is A234741(43), and in binary '101011', encodes polynomial x^5 + x^3 + x + 1, which factors as (x + 1)(x^4 + x^3 + 1), i.e. 43 = A048720(3,25), and the latter factor of these, encoded by 25, is a composite in N, thus 43 is a term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..1605

Setwise difference A236838 \ A236834, and also A236841 \ A236850.

Cf. A236838, A236841, A236849.

cp's OEIS Frontend

A014580 Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A234742 Product of the binary encodings of the irreducible factors (with multiplicity) of the polynomial over GF(2) whose encoding is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A091214 Composite numbers whose binary representation encodes a polynomial irreducible over GF(2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A236837 The greatest inverse of A234741: a(n) = the largest k such that A234741(k) = n, and 0 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

A236851 Remultiply n first "upward", from GF(2)[X] to N, and then remultiply that result back "downward", from N to GF(2)[X]: a(n) = A234741(A234742(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

A236860 After 0 and 1, numbers n all of whose prime divisors encode an irreducible polynomial over GF(2) (are terms of A091206).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A236838 Numbers whose binary representation encodes a polynomial over GF(2) with the property that at least one of its irreducible factors is encoded by a composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

A236839 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number and have at least one irreducible polynomial factor encoded by a composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs