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

%I A110700 #16 Mar 13 2023 10:19:24
%S A110700 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,
%T A110700 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,
%U A110700 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
%N A110700 Number of zeros in the smallest prime with Hamming weight n (given by A061712).
%C A110700 a(n)=0 iff n belongs A000043.
%C A110700 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
%H A110700 T. D. Noe, <a href="/A110700/b110700.txt">Table of n, a(n) for n=1..1024</a>
%F A110700 a(n) = A110699(n) - n.
%p A110700 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;
%t A110700 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 *)
%Y A110700 Cf. A000043, A061712, A110699.
%K A110700 nonn,base
%O A110700 1,6
%A A110700 _Max Alekseyev_, Aug 03 2005