A110700 Number of zeros in the smallest prime with Hamming weight n (given by A061712).

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

Offset: 1

Author

Max Alekseyev, Aug 03 2005

Keywords

• nonn
• base

Comments

a(n)=0 iff n belongs A000043.

Observe that a(n)=3 for n=6, 14, 30, 62, 126, 254, 510, 1022, ... which is A000918. Conjecture: a(n) is never greater than 3. - T. D. Noe, Mar 14 2008

Links

• T. D. Noe, Table of n, a(n) for n=1..1024

Crossrefs

Cf. A000043, A061712, A110699.

Programs

Maple

with(combstruct); a:=proc(n) local m,is,s,t,r; if n=1 then return 1 fi; r:=+infinity; for m from 0 do is := iterstructs(Combination(n-2+m),size=n-2); while not finished(is) do s := nextstruct(is); t := 2^(n-1+m)+1+add(2^i,i=s); if isprime(t) then return m fi; od; od; return 0; end;

Mathematica

A061712[n_] := A061712[n] = Module[{m, s, k, p}, For[m=0, True, m++, s = {1, Sequence @@ #, 1} & /@ Permutations[Join[Table[1, {n - 2}], Table[0, {m}]]] // Sort; For[k=1, k <= Length[s], k++, p = FromDigits[s[[k]], 2]; If[PrimeQ[p], Return[p]]]]]; A061712[1]=2; Table[DigitCount[A061712[n], 2, 0], {n, 1, 100}] (* Jean-François Alcover, Mar 16 2015 *)

Formula

a(n) = A110699(n) - n.