A254503 Möbius transform of A034448.

1, 2, 3, 2, 5, 6, 7, 4, 6, 10, 11, 6, 13, 14, 15, 8, 17, 12, 19, 10, 21, 22, 23, 12, 20, 26, 18, 14, 29, 30, 31, 16, 33, 34, 35, 12, 37, 38, 39, 20, 41, 42, 43, 22, 30, 46, 47, 24, 42, 40, 51, 26, 53, 36, 55, 28, 57, 58, 59, 30, 61, 62, 42, 32, 65, 66, 67, 34, 69, 70

Offset

1,2

Links

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Formula

a(n) = phi(A057521(n)) * A055231(n).

If n is squarefree, a(n) = n; if n is powerful, a(n) = phi(n).

Multiplicative with a(p) = p; a(p^e) = phi(p^e), for e > 1.

Dirichlet g.f.: zeta(s-1) / zeta(2s-1).

a(n) = Sum_{d|n, gcd(n/d, d) = 1} mu(d)^2 * phi(n/d). - Daniel Suteu, Jun 27 2018

Sum_{k=1..n} a(k) ~ n^2 / (2*zeta(3)). - Vaclav Kotesovec, Jan 11 2019

Mathematica

Table[DivisorSum[n, MoebiusMu[#]^2*EulerPhi[n/#] &, CoprimeQ[n/#, #] &], {n, 70}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := (p - 1)*p^(e - 1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1); ); factorback(f); } \\ Michel Marcus, Feb 06 2015
(PARI) a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d))); \\ Daniel Suteu, Jun 27 2018

Crossrefs

Cf. A000010 (totient), A001694 (powerful), A005117 (squarefree), A034448 (usigma), A057521 (powerful part), A055231 (unitary squarefree kernel).

Sequence in context: A019554 A371573 A076685 * A186646 A309108 A308056

Adjacent sequences: A254500 A254501 A254502 * A254504 A254505 A254506

Keyword

mult,nonn,easy

Author

Álvar Ibeas, Jan 31 2015

Status

approved