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 *)

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: A355263 A019554 A076685 * A186646 A309108 A308056

Adjacent sequences: A254500 A254501 A254502 * A254504 A254505 A254506

Keyword

mult,nonn

Author

Álvar Ibeas, Jan 31 2015

Status

approved