OFFSET
1,1
COMMENTS
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
From Antti Karttunen, Jul 03 2024: (Start)
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.
Bower's formula is now submitted as A374053.
(End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..131072 (first 100000 terms from Antti Karttunen)
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.
FORMULA
a(n) = A011655(n) * A353815(n), conjectured by Antti Karttunen, proved by Seiichi Manyama, Jul 03 2024
MATHEMATICA
a[n_] := Boole[!Divisible[RamanujanTau[n], 3]]; Array[a, 92] (* Jean-François Alcover, Jul 05 2017 *)
PROG
(PARI) A070563(n) = !!(ramanujantau(n)%3); \\ Antti Karttunen, Jul 02 2024
(PARI) A070563(n) = ((n%3) && (sigma(n)%3)); \\ Antti Karttunen, Jul 03 2024
(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
CROSSREFS
Characteristic function of A374135, nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3.
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, May 07 2002
STATUS
approved