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A096369 Triangle read by rows, 0<=k T(n,k) = #{p prime: b(k)=1 and 2^(n-1) <= p=Sum(b(i)*2^i:0<=b(i)<2) < 2^n}.

%I A096369 #7 Sep 14 2021 20:50:07
%S A096369 0,1,2,2,1,2,2,1,1,2,5,3,3,2,5,7,3,4,5,3,7,13,7,6,6,4,7,13,23,13,12,9,
%T A096369 10,12,11,23,43,22,23,22,23,22,21,21,43,75,37,37,36,40,39,38,38,37,75,
%U A096369 137,71,71,73,66,56,71,70,66,67,137,255,128,125,130,127,132,128,130,129,126,125,255
%N A096369 Triangle read by rows, 0<=k<n: T(n,k) = #{p prime: b(k)=1 and 2^(n-1) <= p=Sum(b(i)*2^i:0<=b(i)<2) < 2^n}.
%C A096369 T(n,0) = A036378(n-1) for n>1; T(n,n-1) = T(n,0) for n>2;
%C A096369 T(n,1) = A095008(n-1) for n>2;
%C A096369 T(n,n-2) = A095766(n-1) for n>1;
%C A096369 conjecture: T(n,k) > 0 for n>1.
%e A096369 prime(12) = 37 -> 1 0 0 1 0 1 ..... n = 6
%e A096369 prime(13) = 41 -> 1 0 1 0 0 1 ..... all primes p:
%e A096369 prime(14) = 43 -> 1 0 1 0 1 1 ..... 2^(6-1) <= p < 2^6
%e A096369 prime(15) = 47 -> 1 0 1 1 1 1
%e A096369 prime(16) = 53 -> 1 1 0 1 0 1
%e A096369 prime(17) = 59 -> 1 1 1 0 1 1
%e A096369 prime(18) = 61 -> 1 1 1 1 0 1
%e A096369 col-sums of bits: 7 3 5 4 3 7 : T(6,5)=7, T(6,4)=3, T(6,3)=5,
%e A096369 ...
%t A096369 S[n_] := S[n] = IntegerDigits[Select[Range[2^(n-1), 2^n], PrimeQ], 2] // Transpose;
%t A096369 T[1, 1] = 0;
%t A096369 T[n_, k_] := S[n][[n-k+1]] // Total;
%t A096369 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 14 2021 *)
%K A096369 nonn,tabl
%O A096369 1,3
%A A096369 _Reinhard Zumkeller_, Jul 19 2004