A280998 Numbers with a prime number of 1's in their binary reflected Gray code representation.

2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 103

Offset: 1

Author

Indranil Ghosh, Jan 12 2017

Keywords

• nonn
• base
• easy

Comments

From Emeric Deutsch, Jan 28 2018: (Start)

Also the indices of the compositions that have a prime number of parts. For the definition of the index of a composition see A298644.

For example, 27 is in the sequence since its binary form is 11011 and the composition [2,1,2] has 3 parts.

On the other hand, 58 is not in the sequence since its binary form is 111010 and the composition [3,1,1,1] has 4 parts.

The command c(n) from the Maple program yields the composition having index n. (End)

Examples

			27 is in the sequence because the binary reflected Gray code representation of 27 is 10110 which has 3 1's, and 3 is prime.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..10001
• Wikipedia, Gray code.

Crossrefs

Cf. A014550, A052294, A005811, A298644, A101211.

Programs

Maple

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]:
for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1:
r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc:
RunLengths := proc (L) map(nops, Runs(L)) end proc:
c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc:
A := {}: for n to 175 do if isprime(nops(c(n))) = true then A := `union`(A, {n}) else end if end do: A;
# most of the program is due to W. Edwin Clark. # Emeric Deutsch, Jan 28 2018

Mathematica

Select[Range[100], PrimeQ[DigitCount[BitXor[#, Floor[#/2]], 2, 1]] &] (* Amiram Eldar, May 01 2021 *)

PARI

is(n)=isprime(hammingweight(bitxor(n, n>>1))) \\ Charles R Greathouse IV, Jan 12 2017