A354204 a(n) = phi(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

1, 4, 6, 20, 12, 24, 10, 100, 42, 48, 18, 120, 16, 40, 72, 500, 28, 168, 22, 240, 60, 72, 30, 600, 156, 64, 294, 200, 36, 288, 42, 2500, 108, 112, 120, 840, 40, 88, 96, 1200, 52, 240, 46, 360, 504, 120, 58, 3000, 110, 624, 168, 320, 60, 1176, 216, 1000, 132, 144, 66, 1440, 72, 168, 420, 12500, 192, 432, 70, 560, 180

Offset: 1

Author

Antti Karttunen, May 23 2022

Keywords

• nonn
• mult

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Möbius transform of A354202.

Cf. A000010, A008683, A354200, A354205.

Programs

PARI

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354204(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); eulerphi(factorback(f)); };
\\ Alternatively:
A354204v2(n) = { my(f=factor(n),q); prod(k=1,#f~,q = A354200(primepi(f[k,1])); (q-1)*(q^(f[k,2]-1))); };

Formula

Multiplicative with a(p^e) = (q-1) * q^(e-1), where q = A354200(A000720(p)).

a(n) = A000010(A354202(n)).

a(n) = Sum_{d|n} A008683(n/d) * A354202(d).