cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A344143 Indices k such that A344141(k) and A344142(k) are not equal.

Original entry on oeis.org

33, 34, 36, 37, 42, 49, 54, 55, 58, 59, 62, 65, 68, 71, 72, 73, 74, 76, 78, 79, 80, 82, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 98, 100, 102, 103, 106, 107, 108, 110, 111, 113, 115, 118, 121, 124, 125, 126, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141
Offset: 1

Views

Author

Jianing Song, May 10 2021

Keywords

Comments

A344141 and A344142 are two different methods of finding the "first irreducible GF(2)[X] polynomial of degree k". Sequence gives k such that this two methods disagree.
Obviously, k is a term if and only if A000120(A344141(k)) != A000120(A344142(k)).

Examples

			33 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 33 is x^33 + x^6 + x^3 + x + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^33 + x^10 + 1.
37 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 37 is x^37 + x^5 + x^4 + x^3 + x^2 + x + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^37 + x^6 + x^4 + x + 1.
54 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 54 is x^54 + x^6 + x^5 + x^4 + x^3 + x^2 + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^54 + x^9 + 1.

Links

Crossrefs

Cf. A014580, A344141, A344142, A000120.

Programs

  • PARI
    isA344143(n) = my(k=A344142(n)-1); while(k>=2^n, if(polisirreducible(Mod(Pol(binary(k)), 2)), return(1), k--)); 0 \\ See A344142 for its program, assuming that an irreducible polynomial of degree n with at most 5 terms exists for every n.

A344185 a(n) = A344141(n) - 2^n.

Original entry on oeis.org

0, 3, 3, 3, 5, 3, 3, 27, 3, 9, 5, 9, 27, 33, 3, 43, 9, 9, 39, 9, 5, 3, 33, 27, 9, 27, 39, 3, 5, 3, 9, 141, 75, 27, 5, 53, 63, 99, 17, 57, 9, 39, 89, 33, 27, 3, 33, 45, 113, 29, 75, 9, 71, 125, 71, 149, 17, 99, 123, 3, 39, 105, 3, 27, 27, 9, 39, 163, 101, 43
Offset: 1

Views

Author

Jianing Song, May 11 2021

Keywords

Comments

A more intuitive version of A344141.
Every term other than the first is a member of A129771.
In A057496 it is stated that if x^n + x^3 + x^2 + x + 1 is irreducible, then so is x^n + x^3 + 1. It follows that no term can be equal to 15.
It is conjectured that no term can be of the form P_m(2^k), where P_m(x) = Product_{i>=0} (1 + x^(2^(d_i)))^(c_i) if the binary representation of m is m = Sum_{i>=0} c_i * 2^(d_i), k is an odd number. See my conjecture in A344177.

Examples

			See A344141.

Links

Crossrefs

Cf. A014580, A344141, A344186, A129771, A344177.

Programs

  • PARI
    A344185(n) = for(k=0, 2^n-1, if(polisirreducible(Mod(Pol(binary(2^n+k)), 2)), return(k)))

A344142 Lexicographically first irreducible polynomial over GF(2) of degree n with the lowest possible number of terms, evaluated at X = 2.

Original entry on oeis.org

2, 7, 11, 19, 37, 67, 131, 283, 515, 1033, 2053, 4105, 8219, 16417, 32771, 65579, 131081, 262153, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108891, 134217767, 268435459, 536870917, 1073741827, 2147483657, 4294967437, 8589935617
Offset: 1

Views

Author

Jianing Song, May 10 2021

Keywords

Comments

Different from A344141, here you first check x^n + x + 1, x^n + x^2 + 1, ..., x^n + x^(n-1) + 1 until you get an irreducible polynomial over GF(2); if there are none, you then check x^n + x^3 + x^2 + x + 1, x^n + x^4 + x^2 + x + 1, x^n + x^4 + x^3 + x + 1, x^n + x^4 + x^3 + x^2 + 1, ..., x^n + x^(n-1) + x^(n-2) + x^(n-3) + 1 until you get an irreducible polynomial over GF(2). Once you find it, evaluate it at x = 2.
Note that it is conjectured that an irreducible polynomial of degree n with 5 terms exists for every n. It follows from the conjecture that A000120(a(n)) = 3 for n in A073571 and 5 for n in A057486.
In A057496 it is stated that if x^n + x^3 + x^2 + x + 1 is irreducible, then so is x^n + x^3 + 1. It follows that no term other than 19 can be of the form 2^n + 15.

Examples

			a(33) = 8589935617, since x^33 + x + 1, x^33 + x^2 + 1, x^33 + x^3 + 1, ..., x^33 + x^9 + 1 are all reducible over GF(2) and x^33 + x^10 + 1 is irreducible, so a(33) = 2^33 + 2^10 + 1 = 8589935617.
a(8) = 283, since x^8 + x + 1, x^8 + x^2 + 1, ..., x^8 + x^7 + 1 are all reducible over GF(2); both x^8 + x^3 + x^2 + x + 1, x^8 + x^4 + x^2 + x + 1 are reducible, and x^8 + x^4 + x^3 + x + 1 is irreducible, so a(8) = 2^8 + 2^4 + 2^3 + 2 + 1 = 283.

Links

Crossrefs

Cf. A014580, A344141, A344143, A000120, A073571, A057486, A057496.

Programs

  • PARI
    A344142(n) = if(n==1, 2, for(k=1, n-1, if(polisirreducible(Mod(x^n+x^k+1, 2)), return(2^n+2^k+1))); for(a=3, n-1, for(b=2, a-1, for(c=1, b-1, if(polisirreducible(Mod(x^n+x^a+x^b+x^c+1, 2)), return(2^n+2^a+2^b+2^c+1)))))) \\ Assuming that an irreducible polynomial of degree n with at most 5 terms exists for every n.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.