A064179 - OEIS

%I #83 Oct 12 2024 16:37:36

%S 1,-1,-1,-1,-1,1,-1,1,-1,1,-1,1,-1,1,1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,1,

%T 1,-1,-1,-1,1,1,1,1,1,-1,1,1,-1,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,-1,-1,1,

%U -1,1,1,-1,-1,-1,1,1,1,1,-1,-1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,-1,1,1,1,-1,-1,-1,1,1,1,1,1,-1,-1,1,1,1,-1,-1,-1

%N Infinitary version of Moebius function: infinitary MoebiusMu of n, equal to mu(n) iff mu(n) differs from zero, else 1 or -1 depending on whether the sum of the binary digits of the exponents in the prime decomposition of n is even or odd.

%C Apparently the (ordinary) Dirichlet inverse of A050377. - _R. J. Mathar_, Jul 15 2010

%C Also analog of Liouville's function (A008836) in Fermi-Dirac arithmetic, where the terms of A050376 play the role of primes (see examples). - _Vladimir Shevelev_, Oct 28 2013.

%D Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian)

%H Antti Karttunen, <a href="/A064179/b064179.txt">Table of n, a(n) for n = 1..65537</a>

%H G. L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integers' infinitary divisors</a>, Math. Comp. 54 (1990), 395-411.

%H G. L. Cohen and P. Hagis, Jr, <a href="http://dx.doi.org/10.1155/S0161171293000456">Arithmetic functions associated with the infinitary divisors of an integer</a>, Internat. J. Math. Math. Sci. 16 (2) (1993), 373-384.

%H Simon Litsyn and Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

%H Rasa Steuding, Jörn Steuding, and László Tóth, <a href="https://doi.org/10.1007/s12215-011-0022-x">A modified Möbius mu-function</a>, Rendiconti del Circolo Matematico di Palermo, Vol. 60 (2011), pp. 13-21; <a href="https://arxiv.org/abs/1109.4242">arXiv preprint</a>, arXiv:1109.4242 [math.NT], 2011.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%F From _Vladimir Shevelev_ Feb 20 2011: (Start)

%F Sum_{d runs through i-divisors of n} a(d)=1 if n=1, or 0 if n>1; Sum_{d runs through i-divisors of n} a(d)/d = A091732(n)/n.

%F Infinitary Moebius inversion:

%F If Sum_{d runs through i-divisors of n} f(d)=F(n), then f(n) = Sum_{d runs through i-divisors of n} a(d)*F(n/d). (End)

%F a(n) = (-1)^A064547(n). - _R. J. Mathar_, Apr 19 2011

%F Let k=k(n) be the number of terms of A050376 that divide n with odd maximal exponent. Then a(n) = (-1)^k. For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96)=3 and a(96)=-1. - _Vladimir Shevelev_, Oct 28 2013

%F From _Peter Munn_, Jan 25 2020: (Start)

%F a(A050376(n)) = -1; a(A059897(n,k)) = a(n) * a(k).

%F a(n^2) = a(n).

%F a(A003961(n)) = a(n).

%F a(A225546(n)) = a(n).

%F a(A000028(n)) = -1; a(A000379(n)) = 1.

%F (End)

%F a(n) = a(A007913(n)) * a(A008833(n)) = (-1)^A001221(A007913(n)) * a(A000188(n)). - _Peter Munn_, Mar 16 2022

%F From _Amiram Eldar_, Dec 23 2023: (Start)

%F Multiplicative with a(p^e) = (-1)^A000120(e).

%F Dirichlet g.f.: 1/Product_{k>=0} zeta(2^k * s) (Steuding et al., 2011). (End)

%e G.f. = x - x^2 - x^3 - x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 + ...

%e mu[45]=0 but iMoebiusMu[45]=1 because 45 = 3^2 * 5^1 and the binary digits of 2 and 1 add up to 2, an even number.

%e A unique representation of 48 over distinct terms of A050376 is 3*16. Since it contains even factors, then a(48)=1; for 54 such a representation is 2*3*9, thus a(54)=-1. - _Vladimir Shevelev_, Oct 28 2013

%t iMoebiusMu[n_] := Switch[MoebiusMu[n], 1, 1, -1, -1, 0, If[OddQ[Plus@@(DigitCount[Last[Transpose[FactorInteger[n]]], 2, 1])], -1, 1]];

%t (* The Moebius inversion formula seems to hold for iMoebiusMu and the infinitary_divisors of n: if g[ n_ ] := Plus@@(f/@iDivisors[ n ]) for all n, then f[ n_ ]===Plus@@(iMoebiusMu[ # ]g[ n/# ])/@iDivisors[ n ]) *)

%t f[p_, e_] := (-1)^DigitCount[e, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 23 2023 *)

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; (-1) ^ subst( Pol( binary(e)), x, 1)))}; /* _Michael Somos_, Jan 08 2008 */

%o (PARI) a(n) = if (n==1, 1, (-1)^omega(core(n)) * a(core(n,1)[2])) \\ _Peter Munn_, Mar 16 2022

%o (PARI) a(n) = vecprod(apply(x -> (-1)^hammingweight(x), factor(n)[, 2])); \\ _Amiram Eldar_, Dec 23 2023

%o (Scheme) (define (A064179 n) (expt -1 (A064547 n))) ;; _Antti Karttunen_, Nov 23 2017

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A064179(n): return prod(-1 if e.bit_count()&1 else 1 for e in factorint(n).values()) # _Chai Wah Wu_, Oct 12 2024

%Y Sequences with related definitions: A008683, A008836, A064547, A302777.

%Y Positions of -1: A000028.

%Y Positions of 1: A000379.

%Y Cf. A000120, A001221, A050376, A050377, A091732.

%Y Sequences used to express relationships between the terms: A000188, A003961, A007913, A008833, A059897, A225546.

%K sign,mult,easy

%O 1,1

%A _Wouter Meeussen_, Sep 20 2001