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%I #23 Jul 10 2017 04:11:42
%S 1,1,1,-3,1,-5,1,-11,-8,-9,1,-15,1,-13,-14,-7,1,-20,1,-23,-20,-21,1,
%T -5,-24,-25,-26,-31,1,-30,1,41,-32,-33,-34,0,1,-37,-38,-1,1,-40,1,-47,
%U -38,-45,1,65,-48,-44,-50,-55,1,10,-54,3,-56,-57,1,10,1,-61,-50
%N Let f be the multiplicative function satisfying f(p^k) = (1 + p*I)^k for any prime p and k > 0 (where I^2 = -1); a(n) = the real part of f(n).
%C See A289311 for the imaginary part of f.
%C See A289320 for the square of the norm of f.
%C a(p) = 1 for any prime p.
%C If a(n) = 0, then a(n^(2*k-1)) = 0 and A289311(n^(2*k)) = 0 for any k > 0.
%C a(n) = 0 iff Sum_{i=1...k} ( arctan(p_i) * e_i } = Pi/2 * (2*j + 1) for some integer j (where Product_{i=1..k} p_i^e_i is the prime factorization of n).
%C a(n) = 0 for n = 36, 3969, 13608, 46656, 1500282, 5143824, 6718383, ...
%C As a(36) = 0 and 36 = 2^2 * 3^3, we have arctan(2)*2 + arctan(3)*2 = Pi/2 * (2*j + 1) (with j = 1).
%C If |a(n)| = |A289311(n)|, then |a(n^(2k-1))| = |A289311(n^(2k-1))| for any k > 0.
%C |a(n)| = |A289311(n)| iff Sum_{i=1...k} ( arctan(p_i) * e_i } = Pi/4 * (2*j + 1) for some integer j (where Product_{i=1..k} p_i^e_i is the prime factorization of n).
%C |a(n)| = |A289311(n)| for n = 6, 63, 216, 2268, 7776, 23814, 81648, 106641, 250047, 279936, 312273, 857304, ...
%C As |a(63)| = |A289311(63)| and 63 = 3^2 * 7, we have arctan(3)*2 + arctan(7) = Pi/4 * (2*j + 1) (with j=1).
%C The scatterplot of this sequence vs A289311 is interesting (see Links section).
%H Rémy Sigrist, <a href="/A289310/b289310.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A289310/a289310.png">Scatterplot of the first 1000000 terms of A289310 vs A289311</a>
%e f(12) = f(2^2 * 3) = (1 + 2*I)^2 * (1 + 3*I) = -15 - 5*I, hence a(12) = -15.
%t Array[Re[Times @@ Map[(1 + #1 I)^#2 & @@ # &, FactorInteger@ #]] &, 63] (* _Michael De Vlieger_, Jul 03 2017 *)
%o (PARI) a(n) = my (f=factor(n)); real (prod(i=1, #f~, (1 + f[i,1]*I) ^ f[i,2]))
%Y Cf. A289311, A289320.
%K sign
%O 1,4
%A _Rémy Sigrist_, Jul 02 2017
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