|
%I #32 Dec 27 2023 10:26:18
%S 1,1,4,2,4,4,8,4,12,4,12,8,12,8,16,8,16,12,20,8,32,12,24,16,20,12,36,
%T 16,28,16,32,16,48,16,32,24,36,20,48,16,40,32,44,24,48,24,48,32,56,20,
%U 64,24,52,36,48,32,80,28,60,32,60,32,96,32,48,48,68,32
%N Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p = 1 (mod 4) and H(p) = p + 1 if p = 3 (mod 4).
%H Antti Karttunen, <a href="/A204617/b204617.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) = phi(n) if n is in A072437.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - _Amiram Eldar_, Dec 24 2022
%F a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - _Peter Bala_, Dec 26 2023
%p with(numtheory):
%p a := n->add(jacobi(-1,d)*mobius(d)*n/d, d in divisors(n)):
%p seq(a(n), n = 1..60); # _Peter Bala_, Dec 26 2023
%t ar[p_,s_] := Which[Mod[p,4]==1, p^(s-1)*(p-1), Mod[p,4]==3, p^(s-1)*(p+1), True,p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]
%o (PARI) A204617(n) = { my(f=factor(n),p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p,1,if(1==(p%4),p-1,p+1))); }; \\ _Antti Karttunen_, Nov 16 2021
%Y Cf. A000010, A072437.
%Y Cf. A175647, A243381.
%K nonn,mult
%O 1,3
%A _José María Grau Ribas_, Jan 17 2012
|
