Smallest semiprime with Hamming weight n

[Jonathan Post]

Against my original intent, I did just submit the below because: (1)
Stefan Steinerberger not only extended it, but had interesting related
seqs of his own; (2) there were many more cross-references than I
expected; (3) both Hamming distance and semiprimes (for public key
cryptosystems) are important to mathematical coding theory, a subject
in which njas has tremendous expertise.  By the way, I took the coding
theory course at Caltech when Golomb taught it, and it remains a
beautiful and important subject.

NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167,
12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727,
6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303,
805306367, 1879048191, 3221225471,  6442450943, 15032385535,
34225520639, 64424509439, 137371844607, 137438953471, 412316860415
%N A000001 Smallest semiprime with Hamming weight n (i.e. smallest
semiprime with exactly n ones when written in binary).
%C A000001 Semiprime analogue of A061712. Extended by Stefan
Steinerberger (stefan.steinerberger(AT)gmail.com). Includes the subset
Mersenne semiprimes A092561.
%F A000001 a(n) = MIN{k in A001358: A000120(A007088(k)) = n}.
%e A000001 a(1) = 4 because the first semiprime A001358(1) is 4 (base
10) which is written 100 in binary, the latter representation having
exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 one.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones
and, as it also has no zeros, is the smallest of the Mersenne
semiprimes.
%Y A000001 Cf. A000043, A000120, A000337, A000668, A001358, A007088,
A061712, A085724, A089226, A089998, A089999, A091991, A092558,
A092559, A092561, A092562, A093535, A102782, A110472, A110699,
A110700.
%O A000001 1,1
%K A000001 ,base,easy,nonn,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jun 23 2007