A127705 - OEIS

%I #18 Sep 02 2023 10:37:24

%S 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 9,31,16,19,15,23,12,37,17,23,16,41,13,43,20,24,21,47,16,42,20,31,24,

%U 53,18,39,24,35,27,59,16,61,29,36,32,47,21,67,32,43,25,71,24,73

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

%C Previous name: Row sums of A127704.

%C From _Robert Israel_, Dec 29 2016: (Start)

%C a(n) = n if and only if n is in A008578.

%C a(p^j) = p^j - p^(j-1) if p is prime and j >= 2.

%C 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 Robert Israel, <a href="/A127705/b127705.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = mu(n) + Sum_{k|n,k>1} (k+1)*mu(n/k), where mu = A008683. - _Robert Israel_, Dec 29 2016

%p N := 100: # to get a(1)..a(N)

%p A:= Vector(N, numtheory:-mobius):

%p for k from 2 to N do

%p   for j from 1 to floor(N/k) do

%p     A[j*k]:= A[j*k] + (k+1)*numtheory:-mobius(j)

%p od od:

%p convert(A, list); # _Robert Israel_, Dec 29 2016

%o (Python)

%o def A127705_list(n):

%o     upto = n + 1

%o     p = [i for i in range(upto)]

%o     for i in range(2, upto):

%o         for j in range(i + i, upto , i):

%o             p[j] -= p[i]

%o     return p[1::]

%o print(A127705_list(1000))  # _Peter Luschny_, Sep 02 2023

%Y Cf. A008578, A008683, A054525, A127701, A127704.

%K nonn

%O 1,2

%A _Gary W. Adamson_, Jan 24 2007

%E Terms a(11) and beyond from _Robert Israel_, Dec 29 2016

%E New name using a formula of _Robert Israel_ by _Peter Luschny_, Sep 02 2023