cp's OEIS Frontend

A298483 Numbers n, the smallest of three consecutive numbers that share the property mu(n) <> chi(n).

%I A298483 #47 Feb 10 2018 11:50:08
%S A298483 13,25,37,61,73,109,113,117,121,153,157,169,173,181,245,257,273,277,
%T A298483 285,289,297,313,317,325,333,353,361,369,373,385,389,401,405,409,421,
%U A298483 425,457,509,513,525,529,541,601,609,621,637,653,673,677,693,705,709,729,733,761,765,769,777,797,801,805,829,833,841,853
%N A298483 Numbers n, the smallest of three consecutive numbers that share the property mu(n) <> chi(n).
%C A298483 mu and chi share the same property in that they both evaluate to {-1, 0, 1}.
%C A298483 This sequence admits 5 possible outcomes as follows:
%C A298483 - a(n) are of the form 4k + 1, and are either divisible by an odd number of primes, or are nonsquarefree.
%C A298483 - a(n) + 1 are squarefree even numbers.
%C A298483 - a(n) + 2 are of the form 4k + 3, and are either divisible by an even number of primes, or are nonsquarefree.
%C A298483 3 is the largest number of consecutive integers that satisfy the condition mu(n) <> chi(n). Since a(n) + 3  = 4k + 4 = 4(k+1), which is both nonsquarefree and even, then mu(4(k+1))= chi(4(k+1)), and the sequence terminates.
%C A298483 If a(n) is prime then k - 2 is not divisible by 3.
%C A298483 Conjecture: Every prime a(n) has a multiple a(j), with j > n, the result of a multiplication by a number of the form 4k + 1, a multiple a(m) + 1, with m > n, the result of multiplication by a squarefree even number, and lastly a multiple a(k) + 2, with k > n, the result of multiplication by a prime. Example; a(1) = 13, a(8) = 117, a(2) + 1 = 26, and a(3) + 2 = 39.
%C A298483 If a(n) + 1 is a totient then k - 2 is not divisible by 3.
%C A298483 Observation: Of the 72762 triples up to 10^6, only 19 of the middle terms, which are always even, are totients.
%F A298483 0 < min({|mu(a(n))| + |chi(a(n))|, |mu(a(n) + 1)| + |chi(a(n) + 1)|, |mu(a(n) + 2)| + |chi(a(n) + 2)|}).
%e A298483 13 is in the sequence because mu(13)=-1 and chi(13)=1, mu(14)=1 and chi(14)=0, and mu(15)=1 and chi(15)=-1.
%t A298483 With[{nn = 10^3, w = {1, 0, -1, 0}}, Position[Map[UnsameQ @@ # & /@ # &, Partition[Transpose@ {Array[MoebiusMu, nn], Array[w[[Mod[#, 4, 1] ]] &, nn]}, 3, 1]], {True, True, True}]][[All, 1]] (* _Michael De Vlieger_, Jan 28 2018, after _Michael Somos_ at A101455 *)
%o A298483 (PARI) isok(n) = (moebius(n) != kronecker( -4, n)) && (moebius(n+1) != kronecker( -4, n+1)) && (moebius(n+2) != kronecker( -4, n+2)); \\ _Michel Marcus_, Jan 28 2018
%Y A298483 Cf. A298482, A016813, A002144, A101455 (chi), A008683 (mu).
%K A298483 nonn
%O A298483 1,1
%A A298483 _Torlach Rush_, Jan 19 2018