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

1,3

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

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).

Mathematica

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

Prog

(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));

Crossrefs

Möbius transform of A336853.

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

Cf. also A346249, A347098.

Sequence in context: A374456 A340148 A049200 * A164701 A325965 A198540

Adjacent sequences: A347097 A347098 A347099 * A347101 A347102 A347103

Keyword

nonn

Author

Antti Karttunen, Aug 19 2021

Status

approved