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 A070563 #40 Jul 04 2024 18:27:17
%S A070563 1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,
%T A070563 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,
%U A070563 0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0
%N A070563 a(n) = 0 if 3 divides the Ramanujan number tau(n) (A000594(n)), otherwise 1.
%C A070563 Multiplicative because A000594 is. Conjecture: a(3^k) = 0, if p == 1 mod 3, a(p^2k) = 0 and a(p^(2k+1)) = 1, if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. - _Christian G. Bower_, Jun 10 2005
%C A070563 From _Antti Karttunen_, Jul 03 2024: (Start)
%C A070563 The above conjecture is not correct. The first counterexample occurs at n = 2401 = 7^4. My improved conjecture is that this is actually a characteristic function of nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3, that is, having a following multiplicative formula: a(3^k) = 0, if p == 1 mod 3, a(p^e) = 1 if e != 2 (mod 3), otherwise 0, and if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. This conjecture has now been proved correct by _Seiichi Manyama_.
%C A070563 Bower's formula is now submitted as A374053.
%C A070563 (End)
%H A070563 Michael De Vlieger, <a href="/A070563/b070563.txt">Table of n, a(n) for n = 1..131072</a> (first 100000 terms from Antti Karttunen)
%H A070563 P. Moree and H. J. J. te Riele, <a href="http://arXiv.org/abs/math.NT/0204332">The hexagonal versus the square lattice</a>, arXiv:math/0204332 [math.NT], 2002.
%H A070563 P. Moree and H. J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-03-01556-4">The hexagonal versus the square lattice</a>, Math. Comp. 73 (2004), no. 245, 451-473.
%H A070563 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A070563 a(n) = A011655(n) * A353815(n), conjectured by _Antti Karttunen_, proved by _Seiichi Manyama_, Jul 03 2024
%t A070563 a[n_] := Boole[!Divisible[RamanujanTau[n], 3]]; Array[a, 92] (* _Jean-François Alcover_, Jul 05 2017 *)
%o A070563 (PARI) A070563(n) = !!(ramanujantau(n)%3); \\ _Antti Karttunen_, Jul 02 2024
%o A070563 (PARI) A070563(n) = ((n%3) && (sigma(n)%3)); \\ _Antti Karttunen_, Jul 03 2024
%o A070563 (PARI) A070563(n) = { my(f=factor(n)); prod(i=1, #f~, if(3==f[i, 1], 0, 1==(f[i, 1]%3), 2!=(f[i, 2]%3), (1+f[i, 2])%2)); }; \\ _Antti Karttunen_, Jul 03 2024
%Y A070563 Characteristic function of A374135, nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3.
%Y A070563 Cf. A000203, A000594, A011655, A070564, A126825, A353815, A374053.
%K A070563 nonn,mult
%O A070563 1,1
%A A070563 _N. J. A. Sloane_, May 07 2002