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A254503 Möbius transform of A034448. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A019530 A019554 A076685 * A186646 A309108 A308056

Adjacent sequences:  A254500 A254501 A254502 * A254504 A254505 A254506

KEYWORD

mult,nonn

AUTHOR

Álvar Ibeas, Jan 31 2015

STATUS

approved

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Last modified September 16 02:16 EDT 2019. Contains 327088 sequences. (Running on oeis4.)