cp's OEIS Frontend

A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i*2^(i-1) and Sum_{i > 0} c_i*2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i*2^(i-1) with d_i = (Sum_{k divides i} b_k*c_{i/k}) mod 2 for any i > 0.

%I A370049 #12 May 01 2024 11:29:43
%S A370049 0,0,0,0,1,0,0,2,2,0,0,3,8,3,0,0,4,10,10,4,0,0,5,32,9,32,5,0,0,6,34,
%T A370049 36,36,34,6,0,0,7,40,39,256,39,40,7,0,0,8,42,46,260,260,46,42,8,0,0,9,
%U A370049 128,45,288,257,288,45,128,9,0,0,10,130,136,292,294,294,292,136,130,10,0
%N A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i*2^(i-1) and Sum_{i > 0} c_i*2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i*2^(i-1) with d_i = (Sum_{k divides i} b_k*c_{i/k}) mod 2 for any i > 0.
%C A370049 The set of nonnegative integers equipped with A form a commutative monoid.
%F A370049 A(n, k) = A(k, n).
%F A370049 A(m, A(n, k)) = A(A(m, n), k).
%F A370049 A(m XOR n, k) = A(m, k) XOR A(n, k) (where XOR denotes the bitwise XOR operator).
%F A370049 A000120(A(n, 2^k)) = A000120(n).
%F A370049 A(n, 0) = 0.
%F A370049 A(n, 1) = n.
%F A370049 A(n, 2) = A062880(n).
%e A370049 Array A(n, k) begins:
%e A370049   n\k | 0   1    2    3     4     5     6     7      8      9     10
%e A370049   ----+-------------------------------------------------------------
%e A370049     0 | 0   0    0    0     0     0     0     0      0      0      0
%e A370049     1 | 0   1    2    3     4     5     6     7      8      9     10
%e A370049     2 | 0   2    8   10    32    34    40    42    128    130    136
%e A370049     3 | 0   3   10    9    36    39    46    45    136    139    130
%e A370049     4 | 0   4   32   36   256   260   288   292   2048   2052   2080
%e A370049     5 | 0   5   34   39   260   257   294   291   2056   2061   2090
%e A370049     6 | 0   6   40   46   288   294   264   270   2176   2182   2216
%e A370049     7 | 0   7   42   45   292   291   270   265   2184   2191   2210
%e A370049     8 | 0   8  128  136  2048  2056  2176  2184  32768  32776  32896
%e A370049     9 | 0   9  130  139  2052  2061  2182  2191  32776  32769  32906
%e A370049    10 | 0  10  136  130  2080  2090  2216  2210  32896  32906  32776
%o A370049 (PARI) bits(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=2^b[k]=valuation(n,2)); return (b); }
%o A370049 A(n, k) = { my (bn = bits(2*n), bk = bits(2*k), v = 0, e); for (i = 1, #bn, for (j = 1, #bk, e = bn[i] * bk[j] - 1; v = bitxor(v, 2^e););); return (v); }
%Y A370049 Cf. A000120, A062880, A048720, A341288.
%K A370049 nonn,base,tabl
%O A370049 0,8
%A A370049 _Rémy Sigrist_, Apr 30 2024