A354094 a(n) = phi(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and phi is Euler totient function.

1, 4, 2, 20, 10, 8, 6, 100, 6, 40, 16, 40, 12, 24, 20, 500, 22, 24, 18, 200, 12, 64, 28, 200, 110, 48, 18, 120, 40, 80, 30, 2500, 32, 88, 60, 120, 36, 72, 24, 1000, 46, 48, 42, 320, 60, 112, 52, 1000, 42, 440, 44, 240, 58, 72, 160, 600, 36, 160, 70, 400, 60, 120, 36, 12500, 120, 128, 66, 440, 56, 240, 82, 600, 72

Offset: 1

Author

Antti Karttunen, May 17 2022

Keywords

• nonn
• mult

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Möbius transform of A354091.

Cf. A000010, A003627, A007949, A008683, A010872, A074942, A354099.

Cf. also A003972.

Programs

PARI

A354094(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); eulerphi(factorback(f)); };

Formula

Multiplicative with a(p^e) = (q-1) * q^(e-1) where q = A003627(1+n) if p = A003627(n), otherwise q = p.

a(n) = A000010(A354091(n)).

a(n) = Sum_{d|n} A008683(n/d) * A354091(d).

For all n >= 1, A010872(a(n)) = A010872(A000010(n)) = A074942(n).

For all n >= 1, A007949(a(n)) = A007949(A000010(n)) = A354099(n).