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 1-10 of 11 results. Next

A234751 Self-inverse permutation of integers induced by the restriction of blue-code to irreducible polynomials over GF(2): a(n) = A091227(A193231(A014580(n))).

Original entry on oeis.org

2, 1, 3, 5, 4, 6, 8, 7, 12, 14, 13, 9, 11, 10, 17, 18, 15, 16, 20, 19, 21, 23, 22, 41, 39, 40, 38, 37, 34, 33, 35, 36, 30, 29, 31, 32, 28, 27, 25, 26, 24, 43, 42, 47, 46, 45, 44, 49, 48, 51, 50, 52, 54, 53, 55, 71, 69, 70, 68, 65, 64, 67, 66, 61, 60, 63, 62, 59, 57, 58, 56
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2014

Keywords

Comments

This permutation is also induced when A234747 is restricted to primes: a(n) = A000720(A234747(A000040(n))) because of the way A234747 has been defined.
Note that each subsequence a(1)..a(A062692(j)) is closed (i.e., no cycles are leaking out) because blue code (A193231) preserves the degree of polynomials over GF(2) it operates upon.

Links

Crossrefs

Cf. A014580, A062692, A193231, A234747, A234750.

Programs

Formula

a(n) = A091227(A234750(n)) = A091227(A193231(A014580(n))).

A049084 a(n) = pi(n) if n is prime, otherwise 0.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 0, 11, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 17, 0, 18, 0, 0, 0, 0, 0, 19, 0, 0, 0, 20, 0, 21, 0, 0, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 0, 0, 24, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

pi(n) is the prime counting function, A000720.
Equals row sums of triangle A143541. - Gary W. Adamson, Aug 23 2008

Links

Crossrefs

a(n) = A091227(A091202(n)).
Cf. A143541.

Programs

  • Haskell
    import Data.List (unfoldr)
a049084 n = a049084_list !! (fromInteger n - 1)
a049084_list = unfoldr x (1, 1, a000040_list) where
   x (i, z, ps'@(p:ps)) | i == p = Just (z, (i + 1, z + 1, ps))
                        | i /= p = Just (0, (i + 1, z, ps'))
-- Reinhard Zumkeller, Apr 17 2012, Mar 31 2012, Sep 15 2011
  • Maple
    A049084 := proc(n)
    local i;
    if isprime(n) then
        for i from 1 do
            if ithprime(i) = n then
                return i;
            end if;
        end do;
    else
        return 0 ;
    fi;
end proc:
seq(A049084(n),n=1..120) ;
  • Mathematica
    Table[PrimePi[n] * Boole[PrimeQ[n]], {n, 92}] (* Jean-François Alcover, Nov 07 2011, after R. J. Mathar *)
Table[If[PrimeQ[n],PrimePi[n],0],{n,100}] (* Harvey P. Dale, Jan 09 2022 *)
  • PARI
    a(n)=if(isprime(n),primepi(n),0) \\ Charles R Greathouse IV, Jan 08 2013

Formula

a(n) = pi(n)*(pi(n) - pi(n-1)), pi = A000720. - Reinhard Zumkeller, Nov 30 2003
a(n) = A000720(n*A010051(n)). - Labos Elemer, Jan 09 2004
a(n) = A000720(n)*A010051(n). - R. J. Mathar, Mar 01 2011

Extensions

Name clarified by Alonso del Arte, Feb 07 2020 at the suggestion of David A. Corneth

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

Original entry on oeis.org

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

Views

Author

David Petry (petry(AT)accessone.com)

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

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

A091225 Characteristic function of A014580: 1 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

a(n) = A010051(A091203(n)) = A010051(A091205(n)). Partial sums give A091226. Cf. A091227. Complementary to A091247.

Programs

  • PARI
    a(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ Michel Marcus, Nov 11 2017

Extensions

Data section extended up to a(120) by Antti Karttunen, Jan 01 2023

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

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

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)).

Links

Crossrefs

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.

Programs

  • PARI
    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

Formula

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

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

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)).

Links

Crossrefs

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.

Programs

  • PARI
    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

Formula

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)

A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

Keywords

Comments

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).
On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

Links

Crossrefs

Inverse: A091205.
Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.
Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

Programs

  • PARI
    v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes 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).
As a composition of related permutations:
a(n) = A245703(A245822(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]
For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]

A091208 A014580-indices of irreducible GF(2)[X]-polynomials that are also primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 19, 20, 21, 24, 25, 28, 29, 32, 33, 35, 37, 38, 39, 42, 46, 55, 58, 60, 62, 65, 68, 69, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88, 91, 94, 95, 98, 99, 100, 101, 106, 109, 112, 113, 116, 117, 119, 120, 121, 127, 128, 129, 130
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

a(n) = A091227(A091206(n)). Complement of A091215.

A091215 A014580-indices of irreducible GF(2)[X]-polynomials that are composite integers.

Original entry on oeis.org

7, 12, 17, 18, 22, 23, 26, 27, 30, 31, 34, 36, 40, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 59, 61, 63, 64, 66, 67, 70, 71, 72, 73, 74, 75, 76, 84, 86, 89, 90, 92, 93, 96, 97, 102, 103, 104, 105, 107, 108, 110, 111, 114, 115, 118, 122, 123, 124, 125
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

a(n) = A091227(A091214(n)). Complement of A091208.

A091249 A014580-indices of primitive irreducible GF(2)[X]-polynomials.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 48, 49, 50, 51, 52, 53, 56, 58, 61, 64, 65, 68, 70, 73, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Links

Crossrefs

a(n) = A091227(A091250(n)). Complement of A091251.
Showing 1-10 of 11 results. Next

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

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