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A335019 Numbers whose prime exponents generates rotationally symmetric XOR-triangles.

%I A335019 #10 May 24 2020 18:33:30
%S A335019 1,2,4,8,16,32,42,64,70,128,256,390,512,1024,1122,1764,2002,2048,2210,
%T A335019 4096,4900,6270,7854,8192,9450,15470,15750,16384,26460,32768,36366,
%U A335019 54978,58786,65536,66990,73500,74088,86710,108290,123480,131072,152100,162690
%N A335019 Numbers whose prime exponents generates rotationally symmetric XOR-triangles.
%C A335019 For any positive number with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), we build a XOR-triangle with (e_m, ..., e_1) as top row, and having each entry in the subsequent rows be the XOR of the two values above it. This sequence lists integers whose XOR-triangle has 3-fold rotational symmetry. A334990 gives the second row of such XOR-triangles for numbers that are not powers of 2.
%C A335019 This sequence has strong connections with A334556: for any n > 0 and k > 0, A019565(A334556(n))^k belongs to this sequence.
%C A335019 Every power of 2 belongs to the sequence.
%C A335019 If m belongs to this sequence, then m^2 also belongs to this sequence.
%H A335019 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%F A335019 A067029(a(n)) = A071178(a(n)).
%e A335019 The number 15750 = 7^1 * 5^2 * 3^3 * 2^1 yields the following XOR-triangle:
%e A335019      1 2 3 1
%e A335019       3 1 2
%e A335019        2 3
%e A335019         1
%e A335019 As this XOR-triangle has rotational symmetry, 15750 belongs to this sequence.
%o A335019 (PARI) is(n) = {
%o A335019     my (e);
%o A335019     if (n==1, e=[],
%o A335019         my (f=factor(n), m=primepi(f[#f~,1]));
%o A335019         e=vector(m, k, valuation(n, prime(m+1-k)))
%o A335019     );
%o A335019     my (x=e);
%o A335019     for (k=1, #e,
%o A335019         if (e[k]!=x[#x], return (0));
%o A335019         if (x[1]!=e[#e+1-k], return (0));
%o A335019         x=vector(#x-1, k, bitxor(x[k], x[k+1]));
%o A335019     );
%o A335019     return (1);
%o A335019 }
%Y A335019 Cf. A019565, A067029, A071178, A334556, A334990.
%K A335019 nonn,base
%O A335019 1,2
%A A335019 _Rémy Sigrist_, May 19 2020