A037888 -id:A037888 - cp's OEIS frontend

A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3, 1, 0, 3, 2, 4, 1, 0, 2, 2, 1, 2, 1, 3, 1, 0, 2, 2, 1, 2, 1, 3, 3, 2, 1, 0, 2, 2, 1, 3, 1, 0, 3, 2, 4, 2, 1, 3, 1, 0, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3

Offset: 0

Antti Karttunen, Jun 05 2004

Least number of flips of "fibits" (changing either 0 to 1 or 1 to 0 in Zeckendorf-expansion A014417(n)) so that a palindrome is produced.

			The integers 0 and 1 look as '0' and '1' also in Fibonacci-representation,
and being palindromes, a(0) and a(1) = 0.
2 has Fibonacci-representation '10', which needs a flip of other 'fibit',
that it would become a palindrome, thus a(2) = 1. Similarly 3 has representation
'100', so flipping for example the least significant fibit, we get '101',
thus a(3)=1 as well. 7 (= F(3)+F(5)) has representation '1010', which needs
two flips to produce a palindrome, thus a(7)=2. Here F(n) = A000045(n).

a(n) = A037888(A003714(n)). A094202 gives the positions of zeros. Cf. also A095732.

A144477 a(n) = minimal number of 0's that must be changed to 1's in the binary expansion of the n-th prime in order to make it into a palindrome.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, 2, 2, 1, 1, 2, 1, 0, 2, 2, 0, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 3, 1, 0, 2, 2, 3, 1, 1, 2, 2, 2, 3, 0, 1, 3, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 3, 1, 2, 1, 3, 1, 2, 2, 0, 2, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3

Offset: 1

Washington Bomfim, Jan 15 2011, following a suggestion from Joerg Arndt

			a(5) = 1 since prime(5) = 11 = 1011_2 becomes a palindrome if we change the third bit to 0.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Subsequence of A037888.

A144477[n_]:=With[{p=IntegerDigits[Prime[n],2]},HammingDistance[p,Reverse[p]]/2];Array[A144477,100] (* Paolo Xausa, Nov 13 2023 *)

HD(p)=
{
  v=binary(p); H=0; j=#v;
  for(k=1,#v, H+=abs(v[k]-v[j]); j--);
  return(H)
};
for(n=1,100, p=prime(n); an=HD(p)/2; print1(an,", "))

a(n) is half the Hamming distance between the binary expansion of prime(n) and its reversal.

Edited by N. J. A. Sloane, Apr 23 2020 at the suggestion of Harvey P. Dale

cp's OEIS Frontend

A095734 Asymmetricity-index for Zeckendorf-expansion A014417(n) of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A144477 a(n) = minimal number of 0's that must be changed to 1's in the binary expansion of the n-th prime in order to make it into a palindrome.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions