This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331853 #8 Jan 31 2020 16:05:32 %S A331853 1,1,2,2,2,3,2,2,2,3,3,4,2,3,3,3,2,3,3,4,2,3,3,4,2,3,3,5,2,3,3,3,2,3, %T A331853 3,4,3,4,4,5,2,3,3,4,3,4,4,6,2,3,3,5,3,4,4,5,2,3,3,5,3,4,4,4,2,3,3,4, %U A331853 3,4,4,5,2,3,3,4,3,4,4,6,2,3,3,4,3,4,4 %N A331853 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then applying the bitwise AND operator to the numbers represented by the blocks. %H A331853 Rémy Sigrist, <a href="/A331853/a331853.gp.txt">PARI program for A331853</a> %H A331853 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a> %F A331853 a(2^k) = 2 for any k > 0. %F A331853 a(2^k-1) = A008619(k+1) for any k >= 0. %e A331853 For n = 6: %e A331853 - the binary representation of 6 is "110", %e A331853 - we can split it in 4 ways: %e A331853 "110" -> 6 %e A331853 "1" and "10" -> 1 AND 2 = 0 %e A331853 "11" and "0" -> 3 AND 0 = 0 %e A331853 "1" and "1" and "0" -> 1 AND 1 AND 0 = 0 %e A331853 - we have 2 distinct values, %e A331853 - hence a(6) = 2. %o A331853 (PARI) See Links section. %Y A331853 See A331851 for similar sequences. %Y A331853 Cf. A008619. %K A331853 nonn,base %O A331853 0,3 %A A331853 _Rémy Sigrist_, Jan 29 2020