A325641 -id:A325641 - 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

A325643 a(1) = 1; for n > 1, a(n) is the least divisor d > 1 of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 7, 2, 23, 2, 25, 2, 3, 2, 29, 2, 31, 2, 3, 2, 7, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 55, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 69, 2, 71, 2, 73, 2, 3, 2, 77, 2, 79, 2, 81, 2, 83, 2, 5, 2, 87, 2, 89, 2, 91, 2, 31, 2, 5, 2, 97, 2, 3, 2, 101, 2, 103, 2, 3

Offset: 1

Antti Karttunen, May 11 2019

nonn

For n > 1, a(n) is the least divisor d of n that is larger than 1 and for which it holds that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then that polynomial is a divisor of (0,1)-polynomial similarly converted from n, when the division is done over GF(2). See the example.

			For n = 21 = 3*7, 3 is not the answer because X^1 + 1 does not divide X^4 + X^2 + 1 (21 is "10101" in binary) over GF(2). However, the next larger divisor 7 works because X^4 + X^2 + 1 = (X^2 + X^1 + 1)^2 when multiplication is done over GF(2) (note that A048720(7,7) = 21). Thus a(21) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A325559 (fixed points after 1), A325563, A325641, A325642.

A325643(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(d))*Mod(1, 2)); if(0==(p%q), return(d)))));

a(2n) = 2.
For all n >= 1, a(A325559(n)) = A325559(n).
For all n >= 1, n = a(n) * A325641(n) = A048720(a(n), A325642(n)).

A325642 a(1) = 1; for n > 1, a(n) = k for the least divisor d > 1 of n such that A048720(d,k) = n for some k.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 7, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 1, 13, 9, 14, 1, 15, 1, 16, 31, 17, 13, 18, 1, 19, 29, 20, 1, 21, 1, 22, 27, 23, 1, 24, 11, 25, 17, 26, 1, 27, 1, 28, 23, 29, 1, 30, 1, 31, 21, 32, 21, 33, 1, 34, 1, 35, 1, 36, 1, 37, 57, 38, 1, 39, 1, 40, 1, 41, 1, 42, 17, 43, 1, 44, 1, 45, 1, 46, 7, 47, 19

Offset: 1

Antti Karttunen, May 11 2019

nonn

For n > 1, we first find the least divisor d of n that is larger than 1 and for which it holds that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then that polynomial is a divisor of (0,1)-polynomial similarly converted from n, when the division is done over GF(2). a(n) is then the quotient polynomial converted back to decimal via its binary encoding. See the example.

			For n = 9, its least nontrivial divisor is 3, and we find that 3 (in binary "11") corresponds to polynomial X + 1, which in this case is a factor of polynomial X^3 + 1 (corresponding to 9 as 9 is "1001" in binary) as the latter factorizes as (X + 1)(X^2 + X + 1) over GF(2), that is, 9 = A048720(3,7). Thus a(9) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A048720, A325641, A325643.

A325642(n) = if(1==n,n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n,d,if((d>1),my(q = Pol(binary(d))*Mod(1, 2)); if(0==(p%q), return(fromdigits(Vec(lift(p/q)),2))))));

For all n >= 1, A048720(a(n), A325643(n)) = n.

cp's OEIS Frontend

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

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A325643 a(1) = 1; for n > 1, a(n) is the least divisor d > 1 of n such that A048720(d,k) = n for some k.

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A325642 a(1) = 1; for n > 1, a(n) = k for the least divisor d > 1 of n such that A048720(d,k) = n for some k.

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