A390608 - OEIS

%I #30 Feb 15 2026 22:55:34

%S 0,1,2,2,4,4,6,4,6,8,10,8,12,12,16,8,16,12,18,16,24,20,22,16,20,24,18,

%T 24,28,24,30,16,40,32,48,24,36,36,48,32,40,36,42,40,48,44,46,32,42,40,

%U 64,48,52,36,80,48,72,56,58,48,60,60,72,32,96,60,66,64,88,72

%N a(n) = phi(n) * omega(n).

%C a(n) is the number of positive integers which are coprime to n multiplied by the number of distinct primes dividing n.

%C a(n) = phi(n) iff n is in A246655.

%C a(n) = cototient(n) iff n is in A007694.

%C a(n) = sigma(n) iff n is in A073567.

%C a(n) = usigma(n) iff n is in A063795.

%H Ridouane Oudra, <a href="/A390608/b390608.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000010(n) * A001221(n).

%F a(n) = A122411(n) - A116512(n).

%F a(n) = Sum_{d|n} f(d) * phi(n/d), where f(n) = A014963(n) - 1.

%F a(n) = Sum_{d|n, d is a prime power} A057237(d) * phi(n/d).

%F a(n) is neither multiplicative nor additive, but it satisfies the mixed relation:

%F a(n*m) = phi(n) * a(m) + phi(m) * a(n), for all n, m such that gcd(n,m) = 1.

%F Dirichlet g.f.: zeta(s-1)/zeta(s) * Sum_{p prime} (p-1)/(p^s-1).

%e a(30) = phi(30)*omega(30) = 8*3 = 24.

%p with(numtheory): seq(phi(n)*nops(factorset(n)), n=1..120);

%t Table[EulerPhi[n]*PrimeNu[n], {n, 1, 120}]

%o (PARI) a(n) = my(f=factor(n)); eulerphi(f)*omega(f); \\ _Michel Marcus_, Nov 12 2025

%Y Cf. A000010, A001221, A000203, A034448, A051953, A122411, A116512, A057237, A014963.

%Y Cf. A057859, A179179, A062355, A246655, A073567, A063795, A007694.

%K nonn,easy

%O 1,3

%A _Ridouane Oudra_, Nov 12 2025