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A127705 a(n) = mu(n) + Sum_{k|n, k>1} (k+1)*mu(n/k), where mu = A008683.

%I A127705 #18 Sep 02 2023 10:37:24
%S A127705 1,2,3,2,5,1,7,4,6,3,11,4,13,5,7,8,17,6,19,8,11,9,23,8,20,11,18,12,29,
%T A127705 9,31,16,19,15,23,12,37,17,23,16,41,13,43,20,24,21,47,16,42,20,31,24,
%U A127705 53,18,39,24,35,27,59,16,61,29,36,32,47,21,67,32,43,25,71,24,73
%N A127705 a(n) = mu(n) + Sum_{k|n, k>1} (k+1)*mu(n/k), where mu = A008683.
%C A127705 Previous name: Row sums of A127704.
%C A127705 From _Robert Israel_, Dec 29 2016: (Start)
%C A127705 a(n) = n if and only if n is in A008578.
%C A127705 a(p^j) = p^j - p^(j-1) if p is prime and j >= 2.
%C A127705 a(Product_{i=1..k} p_i) = Product_{i=1..k} (p_i-1) - (-1)^k if p_1, ..., p_k are distinct primes. (End)
%H A127705 Robert Israel, <a href="/A127705/b127705.txt">Table of n, a(n) for n = 1..10000</a>
%F A127705 a(n) = mu(n) + Sum_{k|n,k>1} (k+1)*mu(n/k), where mu = A008683. - _Robert Israel_, Dec 29 2016
%p A127705 N := 100: # to get a(1)..a(N)
%p A127705 A:= Vector(N, numtheory:-mobius):
%p A127705 for k from 2 to N do
%p A127705   for j from 1 to floor(N/k) do
%p A127705     A[j*k]:= A[j*k] + (k+1)*numtheory:-mobius(j)
%p A127705 od od:
%p A127705 convert(A, list); # _Robert Israel_, Dec 29 2016
%o A127705 (Python)
%o A127705 def A127705_list(n):
%o A127705     upto = n + 1
%o A127705     p = [i for i in range(upto)]
%o A127705     for i in range(2, upto):
%o A127705         for j in range(i + i, upto , i):
%o A127705             p[j] -= p[i]
%o A127705     return p[1::]
%o A127705 print(A127705_list(1000))  # _Peter Luschny_, Sep 02 2023
%Y A127705 Cf. A008578, A008683, A054525, A127701, A127704.
%K A127705 nonn
%O A127705 1,2
%A A127705 _Gary W. Adamson_, Jan 24 2007
%E A127705 Terms a(11) and beyond from _Robert Israel_, Dec 29 2016
%E A127705 New name using a formula of _Robert Israel_ by _Peter Luschny_, Sep 02 2023