cp's OEIS Frontend

A045763 Number of numbers "unrelated to n": m < n such that m is neither a divisor of n nor relatively prime to n.

%I A045763 #47 Sep 03 2024 00:42:49
%S A045763 0,0,0,0,0,1,0,1,1,3,0,3,0,5,4,4,0,7,0,7,6,9,0,9,3,11,6,11,0,15,0,11,
%T A045763 10,15,8,16,0,17,12,17,0,23,0,19,16,21,0,23,5,25,16,23,0,29,12,25,18,
%U A045763 27,0,33,0,29,22,26,14,39,0,31,22,39,0,37,0,35,30,35,14,47,0,39,23,39,0,49
%N A045763 Number of numbers "unrelated to n": m < n such that m is neither a divisor of n nor relatively prime to n.
%C A045763 Suggested by _Wouter Meeussen_.
%C A045763 a(n) = 0 iff n is a prime or 1 or 4. - _Robert G. Wilson v_, Nov 02 2005
%C A045763 From _Farideh Firoozbakht_, Dec 23 2014: (Start)
%C A045763 1. a(p^k) = p^(k-1) - k where p is a prime and k is a positive integer. Hence if p is prime then a(p) = 0 which is a result of the previous comment.
%C A045763 2. If n = 2*p or n = 4*p and p is an odd prime then a(n) = phi(n) - 1.
%C A045763 3. If n = 3*p where p is a prime not equal to 3 then a(n) = (1/2)*phi(n). (End)
%H A045763 Michael De Vlieger, <a href="/A045763/b045763.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A045763 Martin Beumer, <a href="http://www.jstor.org/stable/2310778">The Arithmetical Function tau_k(N)</a>, Amer. Math. Monthly, 69, Oct 1962, p. 780 (a(n)=ksi(N)).
%F A045763 a(n) = n + 1 - d(n) - phi(n), where d(n) is the number of divisors of n and phi is Euler's totient function.
%F A045763 Dirichlet generating function: zeta(s-1) + zeta(s) - zeta(s)^2 - zeta(s-1)/zeta(s). - _Robert Israel_, Dec 23 2014
%F A045763 a(n) = Sum_{k=1..n} (1 - floor(1/gcd(n,k))) * (ceiling(n/k) - floor(n/k)). - _Wesley Ivan Hurt_, Jan 06 2024
%p A045763 A045763 := proc(n)
%p A045763     n+1-numtheory[tau](n)-numtheory[phi](n) ;
%p A045763 end proc:
%p A045763 seq(A045763(n),n=1..100);# _Robert Israel_, Dec 23 2014
%t A045763 f[n_] := n + 1 - DivisorSigma[0, n] - EulerPhi[n]; Array[f, 84] (* _Robert G. Wilson v_ *)
%o A045763 (PARI) a(n)=n+1-numdiv(n)-eulerphi(n) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A045763 (Python)
%o A045763 from sympy import divisor_count, totient
%o A045763 def A045763(n): return n+1-divisor_count(n)-totient(n) # _Chai Wah Wu_, Sep 02 2024
%Y A045763 Cf. A000005, A000010, A133995.
%K A045763 nonn,look
%O A045763 1,10
%A A045763 _N. J. A. Sloane_
%E A045763 More terms from _Robert G. Wilson v_, Nov 02 2005