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

13, 25, 37, 61, 73, 109, 113, 117, 121, 153, 157, 169, 173, 181, 245, 257, 273, 277, 285, 289, 297, 313, 317, 325, 333, 353, 361, 369, 373, 385, 389, 401, 405, 409, 421, 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

Offset: 1

Author

Torlach Rush, Jan 19 2018

Keywords

• nonn

Comments

mu and chi share the same property in that they both evaluate to {-1, 0, 1}.

This sequence admits 5 possible outcomes as follows:

- a(n) are of the form 4k + 1, and are either divisible by an odd number of primes, or are nonsquarefree.

- a(n) + 1 are squarefree even numbers.

- a(n) + 2 are of the form 4k + 3, and are either divisible by an even number of primes, or are nonsquarefree.

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.

If a(n) is prime then k - 2 is not divisible by 3.

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.

If a(n) + 1 is a totient then k - 2 is not divisible by 3.

Observation: Of the 72762 triples up to 10^6, only 19 of the middle terms, which are always even, are totients.

Examples

			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.
		

Crossrefs

Cf. A298482, A016813, A002144, A101455 (chi), A008683 (mu).

Programs

Mathematica

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 *)

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

Formula

0 < min({|mu(a(n))| + |chi(a(n))|, |mu(a(n) + 1)| + |chi(a(n) + 1)|, |mu(a(n) + 2)| + |chi(a(n) + 2)|}).