A278239 -id:A278239 - 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

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).

A278238 a(n) = A278233(A000695(n)) = A278233(n)^2.

Original entry on oeis.org

1, 4, 4, 16, 16, 36, 4, 64, 36, 144, 4, 144, 4, 36, 64, 256, 256, 900, 4, 1296, 16, 36, 36, 576, 4, 36, 144, 144, 36, 576, 4, 1024, 36, 2304, 36, 3600, 4, 36, 144, 5184, 4, 144, 36, 144, 576, 900, 4, 2304, 36, 36, 1024, 144, 36, 3600, 4, 576, 144, 900, 4, 5184, 4, 36, 144

Offset: 1

Antti Karttunen, Nov 17 2016

nonn

Cf. A048720, A000695, A278233, A278239, A278254.

(define (A278238 n) (A278233 (A000695 n)))

a(n) = A278233(A000695(n)).

a(n) = A278233(n)^2.

A286376 a(n) = A278222(A277699(n)).

Original entry on oeis.org

2, 2, 6, 2, 24, 6, 12, 2, 60, 24, 180, 6, 120, 12, 24, 2, 60, 60, 420, 24, 180, 180, 60, 6, 192, 120, 384, 12, 240, 24, 48, 2, 60, 60, 420, 60, 720, 420, 360, 24, 420, 180, 1260, 180, 4620, 60, 420, 6, 720, 192, 5040, 120, 5040, 384, 1680, 12, 1440, 240, 4320, 24, 480, 48, 96, 2, 60, 60, 420, 60, 2520, 420, 1260, 60, 420, 720, 1260, 420, 1260, 360, 180, 24

Offset: 1

Antti Karttunen, May 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A277699, A278222, A278239, A286243, A286374, A286375.

(define (A286376 n) (A278222 (A277699 n)))

a(n) = A278222(A277699(n)).

cp's OEIS Frontend

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

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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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A278238 a(n) = A278233(A000695(n)) = A278233(n)^2.

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A286376 a(n) = A278222(A277699(n)).

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