A145341 -id:A145341 - cp's OEIS frontend

A145342 a(n) = (A145341(n) + 1)/2.

1, 2, 3, 4, 5, 7, 6, 8, 9, 13, 11, 15, 10, 14, 12, 16, 17, 25, 21, 29, 19, 27, 23, 31, 18, 26, 22, 30, 20, 28, 24, 32, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 34, 50, 42, 58, 38, 54, 46, 62, 36, 52, 44, 60, 40, 56, 48, 64, 65, 97, 81, 113, 73, 105, 89

Offset: 1

Leroy Quet, Oct 08 2008

This sequence is a permutation of the positive integers. It is its own inverse permutation.
Fixed points of the permutation are the terms of A044051. - Ivan Neretin, Oct 31 2015
From Yosu Yurramendi, Feb 04 2019: (Start)
If the terms (n > 0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...
   1;
   2,  3;
   4,  5,  7,  6;
   8,  9, 13, 11, 15, 10, 14, 12;
  16, 17, 25, 21, 29, 19, 27, 23, 31, 18, 26, 22, 30, 20, 28, 24;
  32, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 34, ...
then the following relationship can be observed:
a(1) = 1, a(2) = 2, a(3) = 3,
for m > 0, a(2^(m+1)) = 2*a(2^m), a(2^m + 1) = a(2^m) + 1, a(2^(m+1)+ 2^m) = 2*a(2^(m+1)) - 1, for 0 < k < 2^m, a(2^(m+1)+ k) = 2*a(2^m + k) - 1, a(2^(m+1)+ 2^m + k) = a(2^(m+1) + k) + 1
(End)

Cf. A044051, A145341.

Table[(FromDigits[Reverse[IntegerDigits[2n-1, 2]], 2] +1)/2, {n, 71}] (* Ivan Neretin, Oct 31 2015 *)

a(n) = (1+fromdigits(Vecrev(binary(2*n-1)), 2))/2; \\ Michel Marcus, Feb 04 2019

nmax <- 10^3 # by choice
b <- vector()
for (o in seq(1,nmax,2)){
  w <- which(as.numeric(intToBits(o))==1)
  b <- c(b, sum(2^(max(w)-w)))
}
a <- (b+1)/2
a[1:71]
# Yosu Yurramendi, Feb 04 2019

More terms from R. J. Mathar and Ray Chandler, Oct 10 2008

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

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

A333415 Odd positive integers in base 2 read backwards.

Original entry on oeis.org

1, 11, 101, 111, 1001, 1101, 1011, 1111, 10001, 11001, 10101, 11101, 10011, 11011, 10111, 11111, 100001, 110001, 101001, 111001, 100101, 110101, 101101, 111101, 100011, 110011, 101011, 111011, 100111, 110111, 101111, 111111, 1000001, 1100001, 1010001, 1110001, 1001001

Offset: 1

Devansh Singh, Mar 20 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007088, A145341.

Cf. A004086, A099821.

f:= proc(n) local L,i;
 L:= convert(n,base,2);
 add(L[-i]*10^(i-1),i=1..nops(L));
end proc
map(f, [seq(i,i=1..100,2)]); # Robert Israel, May 19 2024

Table[FromDigits[Reverse[IntegerDigits[n, 2]]], {n, 1, 75, 2}] (* Amiram Eldar, Apr 27 2020 *)

a(n) = fromdigits(Vecrev(binary(2*n-1))); \\ Michel Marcus, May 19 2024

From Michel Marcus, Apr 23 2020: (Start)
a(n) = A007088(A145341(n)).
a(n) = A004086(A099821(n)). (End)

New name, a(1)=1 and more terms from Michel Marcus, Apr 23 2020

cp's OEIS Frontend

A145342 a(n) = (A145341(n) + 1)/2.

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

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A333415 Odd positive integers in base 2 read backwards.

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