A085730 Euler's totient function applied to the sequence of prime powers.

1, 1, 2, 2, 4, 6, 4, 6, 10, 12, 8, 16, 18, 22, 20, 18, 28, 30, 16, 36, 40, 42, 46, 42, 52, 58, 60, 32, 66, 70, 72, 78, 54, 82, 88, 96, 100, 102, 106, 108, 112, 110, 100, 126, 64, 130, 136, 138, 148, 150, 156, 162, 166, 156, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Author

Reinhard Zumkeller, Jul 20 2003

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Power.
• Eric Weisstein's World of Mathematics, Totient Function.

Crossrefs

Cf. A000010, A000961, A025473, A025474, A207193.

Programs

Haskell

a085730 1 = 1
a085730 n = (p - 1) * p ^ (e - 1)
   where p =  a025473 n; e =  a025474 n
-- Reinhard Zumkeller, Feb 16 2012

Mathematica

f[p_, e_] := (p-1)*p^(e-1); s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 220] (* Amiram Eldar, Apr 05 2025 *)

PARI

list(lim)=my(v=List(primes(primepi(lim)))); listput(v,1); for(e=2, log(lim+.5)\log(2),forprime(p=2,(lim+.5)^(1/e),listput(v, p^e))); apply(n->eulerphi(n),vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 30 2012

Formula

a(n) = A000010(A000961(n)).

a(p^e) = (p-1)*p^(e-1).

a(n) = (A025473(n)-1)*A025473(n)^(A025474(n)-1).