A226177 - OEIS

%I #42 Jun 22 2024 20:02:13

%S 1,-2,-2,0,-2,4,-2,0,0,4,-2,0,-2,4,4,0,-2,0,-2,0,4,4,-2,0,0,4,0,0,-2,

%T -8,-2,0,4,4,4,0,-2,4,4,0,-2,-8,-2,0,0,4,-2,0,0,0,4,0,-2,0,4,0,4,4,-2,

%U 0,-2,4,0,0,4,-8,-2,0,4,-8,-2,0,-2,4,0,0,4,-8,-2,0,0,4,-2,0,4,4,4,0,-2,0,4,0,4,4,4,0,-2,0,0,0,-2,-8,-2,0,-8

%N a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

%C The prime numbers are the only solutions to mu(n)*d(n) = -2.

%C Multiplicative with a(p) = -2, a(p^e) = 0, e > 1.

%C The Moebius inverse is A076479, and the Dirichlet inverse A061142. - _R. J. Mathar_, Jun 03 2013

%C Möbius transform of (-1)^omega(n). - _Wesley Ivan Hurt_, Jun 22 2024

%H Antti Karttunen, <a href="/A226177/b226177.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = mu(n)*d(n) = A008683(n)*A000005(n).

%F Sum_{n>0} a(n)/n^s = Product_{p prime} (1 - 2p^(-s)). - _Ralf Stephan_, Jul 07 2013

%F a(n) = mu(n) * 2^omega(n) = |mu(n)| * (-2)^omega(n), where omega = A001221. - _Álvar Ibeas_, Dec 30 2018

%F a(n) = Sum_{d|n} (-1)^omega(d) * mu(n/d). - _Wesley Ivan Hurt_, Jun 22 2024

%e a(5) = mu(5)*d(5) = (-1)(2) = -2.

%p with(numtheory); a:=n->mobius(n)*tau(n); seq(a(k),k=1..100);

%t Table[MoebiusMu[n] DivisorSigma[0, n], {n, 105}] (* _Michael De Vlieger_, Jul 23 2017 *)

%o (PARI) A226177(n) = moebius(n)*numdiv(n); \\ _Antti Karttunen_, Jul 23 2017

%o (Scheme) (define (A226177 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) -2 0) (A226177 (A028234 n))))) ;; _Antti Karttunen_, Jul 23 2017

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X))[n], ", ")); \\ _Vaclav Kotesovec_, Aug 21 2021

%Y Cf. A000005, A000040, A001358, A008683, A074823 (absolute values), A001221.

%K sign,mult

%O 1,2

%A _Wesley Ivan Hurt_, May 29 2013

%E More terms from _Antti Karttunen_, Jul 23 2017

%E Name changed by _David A. Corneth_, Jul 23 2017