A349754 a(n) = phi(A003961(n)) - 2*phi(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and phi is Euler totient function.

-1, 0, 0, 2, -2, 4, -2, 10, 8, 4, -8, 16, -8, 8, 8, 38, -14, 28, -14, 20, 16, 4, -16, 56, 2, 8, 64, 36, -26, 32, -24, 130, 8, 4, 12, 96, -32, 8, 16, 76, -38, 56, -38, 32, 72, 12, -40, 184, 26, 44, 8, 48, -46, 164, -8, 132, 16, 4, -56, 112, -54, 12, 128, 422, 0, 56, -62, 44, 24, 72, -68, 312, -66, 8, 88, 60, 0, 80

Offset: 1

Author

Antti Karttunen, Dec 01 2021

Keywords

• sign

Comments

Möbius transform of A252748.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000010, A003961, A003972, A008683, A083254, A252748, A337549, A347100.

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := EulerPhi[s[n]] - 2*EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 04 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));

Formula

a(n) = A003972(n) - 2*A000010(n) = A337549(n) - A083254(n).

a(n) = A347100(n) - A000010(n).

a(n) = Sum_{d|n} A008683(n/d) * A252748(d).