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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
-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
1,4
COMMENTS
Möbius transform of A252748.
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).
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 *)
PROG
(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));
KEYWORD
sign
AUTHOR
Antti Karttunen, Dec 01 2021
STATUS
approved