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A344841 a(n) is the least positive number not of the form a(k) XOR ... XOR a(m) with 1 <= k <= m < n (where XOR denotes the bitwise XOR operator).

%I A344841 #16 Sep 02 2025 09:22:19
%S A344841 1,2,4,5,8,12,14,16,17,19,20,21,23,32,33,35,48,49,51,56,57,59,64,65,
%T A344841 67,68,69,71,76,77,79,80,81,83,84,85,87,92,93,95,128,129,131,132,133,
%U A344841 135,140,141,143,192,193,195,196,197,199,204,205,207,224,225,227
%N A344841 a(n) is the least positive number not of the form a(k) XOR ... XOR a(m) with 1 <= k <= m < n (where XOR denotes the bitwise XOR operator).
%C A344841 This sequence has similarities with A002048; here we use XOR, there addition.
%C A344841 All powers of 2 appear.
%H A344841 Rémy Sigrist, <a href="/A344841/b344841.txt">Table of n, a(n) for n = 1..10000</a>
%F A344841 Apparently, a(A246360(k)) = 2^(k-1) for any k > 0.
%e A344841   n   a(n)  a(k) XOR ... XOR a(n) for k=1..n
%e A344841   --  ----  -------------------------------------
%e A344841    1     1  1
%e A344841    2     2  3, 2
%e A344841    3     4  7, 6, 4
%e A344841    4     5  2, 3, 1, 5
%e A344841    5     8  10, 11, 9, 13, 8
%e A344841    6    12  6, 7, 5, 1, 4, 12
%e A344841    7    14  8, 9, 11, 15, 10, 2, 14
%e A344841    8    16  24, 25, 27, 31, 26, 18, 30, 16
%e A344841    9    17  9, 8, 10, 14, 11, 3, 15, 1, 17
%e A344841   10    19  26, 27, 25, 29, 24, 16, 28, 18, 2, 19
%o A344841 (PARI) s=2^0; for (n=1, #a=vector(61), print1 (a[n]=valuation(s+1, 2)", "); z=0; forstep (k=n, 1, -1, s=bitor(s, 2^z=bitxor(z,a[k]))))
%o A344841 (Python)
%o A344841 from operator import xor
%o A344841 from itertools import count, accumulate, islice
%o A344841 from collections import deque
%o A344841 def A344841_gen(): # generator of terms
%o A344841     aset, alist = set(), deque()
%o A344841     for k in count(1):
%o A344841         if k in aset:
%o A344841             aset.remove(k)
%o A344841         else:
%o A344841             yield k
%o A344841             aset |= set(k^d for d in accumulate(alist,xor))
%o A344841             alist.appendleft(k)
%o A344841 A344841_list = list(islice(A344841_gen(),60)) # _Chai Wah Wu_, Sep 01 2025
%Y A344841 Cf. A002048, A246360.
%K A344841 nonn,base,changed
%O A344841 1,2
%A A344841 _Rémy Sigrist_, May 29 2021