A049690 a(n) = Sum_{k=1..n} phi(2*k), where phi = Euler totient function, cf. A000010.

0, 1, 3, 5, 9, 13, 17, 23, 31, 37, 45, 55, 63, 75, 87, 95, 111, 127, 139, 157, 173, 185, 205, 227, 243, 263, 287, 305, 329, 357, 373, 403, 435, 455, 487, 511, 535, 571, 607, 631, 663, 703, 727, 769, 809, 833, 877, 923, 955, 997, 1037, 1069, 1117, 1169, 1205

Offset

0,3

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Formula

a(n) ~ 4*n^2/Pi^2. - Vaclav Kotesovec, Aug 20 2021

a(n) = A002088(n) + a(floor(n/2)). - Chai Wah Wu, Aug 04 2024

Maple

A049690 := proc(n) return add(numtheory[phi](2*k), k=1..n): end: seq(A049690(n), n=0..54); # Nathaniel Johnston, May 24 2011

Mathematica

A049690[0]:=0; A049690[n_]:=A049690[n-1]+EulerPhi[2n]; Array[A049690, 200, 0] (* Enrique Pérez Herrero, Feb 25 2012 *)

Prog

(PARI) a(n)=sum(k=1, n, eulerphi(2*k)) \\ Charles R Greathouse IV, Feb 19 2013
(Python)
from sympy import totient
def A049690(n): return sum(totient(n) for n in range(1, n+1, 2)) + (sum(totient(n) for n in range(2, n+1, 2))<<1) # Chai Wah Wu, Aug 04 2024
(Python)
# faster program using program from A002088 and recursive formula
def A049690(n): return A002088(n) + A049690(n>>1) if n else 0 # Chai Wah Wu, Aug 04 2024

Crossrefs

a(n)=b(2n), where b=A049689. Bisections: A099958, A190815.

Cf. A062570.

Sequence in context: A185170 A211340 A061571 * A080075 A007664 A215812

Adjacent sequences: A049687 A049688 A049689 * A049691 A049692 A049693

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Vladeta Jovovic, May 18 2001

Status

approved