A072107 a(n) = Sum_{k=1..n} A014963(k).

1, 3, 6, 8, 13, 14, 21, 23, 26, 27, 38, 39, 52, 53, 54, 56, 73, 74, 93, 94, 95, 96, 119, 120, 125, 126, 129, 130, 159, 160, 191, 193, 194, 195, 196, 197, 234, 235, 236, 237, 278, 279, 322, 323, 324, 325, 372, 373, 380, 381, 382, 383, 436, 437, 438, 439, 440, 441

Offset

1,2

Comments

Is there an expression for lim_{n -> infinity} a(n)/n^2?

Equals row sums of triangle A140582. - Gary W. Adamson, May 17 2008

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Plot of a(n) / (n^2/log(n)) for n = 1..10^9

Formula

a(n) ~ n^2/(2*log(n)). - Vaclav Kotesovec, Jan 30 2025

a(n) = n - A025528(n) + Sum_{k=1..floor(log_2(n))} A034387(floor(n^(1/k))). - Daniel Suteu, Jun 16 2026

Mathematica

Accumulate[Table[Exp[MangoldtLambda[n]], {n, 1, 60}]] (* Amiram Eldar, May 05 2022 *)

Prog

(PARI) a(n)=1+sum(k=2, n, if(if(omega(k)-1, 0, 1)*component(component(factor(k), 1), 1), if(omega(k)-1, 0, 1)*component(component(factor(k), 1), 1), 1))
(Python)
from sympy import primerange
def A072107(n):
    c, k = n, n.bit_length()-1
    for p in primerange(n+1):
        while p**k > n:
            k -= 1
        c += (p-1)*k
    return c # Chai Wah Wu, Jun 17 2026

Crossrefs

Cf. A140582, A014963.

Sequence in context: A350928 A049827 A186736 * A084578 A190493 A070881

Adjacent sequences: A072104 A072105 A072106 * A072108 A072109 A072110

Keyword

easy,nonn,changed

Author

Benoit Cloitre, Jun 19 2002

Status

approved