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 A254503 Möbius transform of A034448. 4
 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 *) 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

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Last modified August 7 01:21 EDT 2024. Contains 375002 sequences. (Running on oeis4.)