A154809 -id:A154809 - cp's OEIS frontend

A207039 Primes whose binary expansion is not palindromic.

2, 11, 13, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317

Offset: 1

Omar E. Pol, Feb 25 2012

Intersection of A000040 and A154809.

Primes in A154809.

Cf. A000040, A006995, A016041.

Select[Table[Prime[n], {n, 200}], IntegerDigits[#, 2] != Reverse[IntegerDigits[#, 2]] &](* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A206922 Root of the n-th binary palindrome. Least number r > 1 such that A006995(n) can be represented by a finite or infinite number of iterations A006995(A006995(A006995(...(...(r))...).

Original entry on oeis.org

2, 2, 3, 4, 4, 6, 4, 8, 6, 10, 11, 12, 13, 14, 4, 16, 8, 18, 19, 20, 6, 22, 23, 24, 25, 26, 10, 28, 29, 30, 11, 32, 12, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 13, 46, 47, 48, 49, 50, 14, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 4, 64, 16, 66, 67, 68

Offset: 1

Hieronymus Fischer, Mar 12 2012

If n is not a binary palindrome, then a(n)=n.

For n>3: a(n)

For n<>3: The number of iterations such that A006995(n)= A006995(A006995(A006995(...(...(r))...) is given by A206921(n).

			a(1)=2, since A006995(1) = 0 = A006995(A006995(2)).
a(3)=3, since A006995(3) = 3 = A006995(A006995(A006995(...(3)...).
a(7)=4, since A006995(7) = 15 = A006995(A006995(A006995(4)).
a(9)=6, since A006995(9) = 21 = A006995(A006995(6)).

Cf. A006995, A206921, A178225, A206915, A154809.

/* C program fragment, omitting formal details, n!=3 */
k=0;
p=A006995(n);
while A178225(p)==1
{
  k++;
  p=A206915(p);
}
return p;

a(n) <= n for n > 1.
a(n)=p(k), where p(k) can be determined by the following iteration: set k=0, p(0)=A006995(n). Repeat while A178225(p(k))==1, set k=k+1, p(k)=A206915(p(k-1)) end repeat [for n<>3].
Recursion for n<>3:
  Case 1: a(n)=n, if n is not a binary palindrome;
  Case 2: a(n)=a(A206915(n)), else.
Formally: a(n)=if (A178225(n)==0) then n else a(A206915(n)).

A330720 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of nonpalindromes.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 1, 1, 2, 2, 0, 1, 1, 1, 0, 1, 3, 3, 3, 3, 3, 3, 2, 2, 3, 4, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 4, 4, 4, 3, 5, 5, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 5, 4, 4, 4, 0, 4

Offset: 0

Rémy Sigrist, Dec 28 2019

This sequence is a variant of A215244.

			For n = 41:
- the binary expansion of 41 is "101001",
- the possible products of nonpalindromes are "101001", "1010"."01", and "10"."10"."01",
- hence a(41) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A020988, A154809, A215244, A301453.

ispali:= proc(L) L = ListTools:-Reverse(L) end proc:
g:= proc(L) option remember; local m;
    add(procname(L[m+1..-1]), m= remove(t -> ispali(L[1..t]),[$1..nops(L)]))
end proc:
g([]):= 1:
seq(g(convert(n,base,2)),n=0..100); # Robert Israel, Dec 29 2019

a(n) = my (b=binary(n), v=b!=Vecrev(b)); for (s=1, #b, my (z=b[s..#b]); if (z!=Vecrev(z), v+=a(fromdigits(b[1..s-1],2)))); v

a(2^k-1) = 0 for any k >= 0.

a(A020988(k+1)) = 2^k for any k >= 0.

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A207039 Primes whose binary expansion is not palindromic.

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A206922 Root of the n-th binary palindrome. Least number r > 1 such that A006995(n) can be represented by a finite or infinite number of iterations A006995(A006995(A006995(...(...(r))...).

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A330720 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of nonpalindromes.

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