A347100 a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.

0, 1, 2, 4, 2, 6, 4, 14, 14, 8, 2, 20, 4, 14, 16, 46, 2, 34, 4, 28, 28, 14, 6, 64, 22, 20, 82, 48, 2, 40, 6, 146, 28, 20, 36, 108, 4, 26, 40, 92, 2, 68, 4, 52, 96, 34, 6, 200, 68, 64, 40, 72, 6, 182, 32, 156, 52, 32, 2, 128, 6, 42, 164, 454, 48, 76, 4, 76, 68, 96, 2, 336, 6, 44, 128, 96, 60, 104, 4, 292, 446, 44, 6

Offset: 1

Author

Antti Karttunen, Aug 19 2021

Keywords

• nonn

Comments

Möbius transform of A336853.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Möbius transform of A336853.

Cf. A000010, A000040, A001223, A003961, A003972, A008683, A051953, A337549.

Cf. also A346249, A347098.

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; a[n_] := EulerPhi[Times @@ f @@@ FactorInteger[n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

PARI

A347100(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (eulerphi(factorback(f))-eulerphi(n)); };

PARI

A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
A347100(n) = sumdiv(n,d,moebius(n/d)*A336853(d));

Formula

a(n) = A003972(n) - A000010(n).

a(n) = A337549(n) + A051953(n).

a(n) = Sum_{d|n} A008683(n/d) * A336853(d).

For all n >= 1, a(A000040(n)) = A001223(n).