A064609 Partial sums of A034448: sum of unitary divisors from 1 to n.

1, 4, 8, 13, 19, 31, 39, 48, 58, 76, 88, 108, 122, 146, 170, 187, 205, 235, 255, 285, 317, 353, 377, 413, 439, 481, 509, 549, 579, 651, 683, 716, 764, 818, 866, 916, 954, 1014, 1070, 1124, 1166, 1262, 1306, 1366, 1426, 1498, 1546, 1614, 1664, 1742, 1814, 1884

Offset

1,2

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Formula

a(n) = a(n-1) + A034448(n) = Sum_{j=1..n} usigma(j) where usigma(j) = A034448(j).

a(n) ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, Jan 11 2019

a(n) = Sum_{k=1..floor(sqrt(n))} k * A008683(k) * A024916(floor(n/k^2)). - Daniel Suteu, Jun 23 2026

Mathematica

Accumulate@ Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 52}] (* Michael De Vlieger, Mar 18 2017 *)

Prog

(PARI) usigma(n)= { local(f, s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ a=0; for (n=1, 1000, a+=usigma(n); write("b064609.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009
(Python)
from sympy.ntheory.factor_ import udivisor_sigma
def a(n): return sum(udivisor_sigma(j, 1) for j in range(1, n + 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017
(Python)
# uses Python code from A068340 and A024916
def A064609(n):
    c, j, w = 0, 1, 0
    while (j2:=j**2) <= n:
        k = n//j2
        m = isqrt(n//k)
        c += (-w+(w:=A068340(m)))*A024916(k)
        j = m+1
    return c # Chai Wah Wu, Jun 23 2026

Crossrefs

Cf. A034448, A064611, A008683, A024916, A068340.

Sequence in context: A365700 A362290 A183865 * A327566 A307159 A365697

Adjacent sequences: A064606 A064607 A064608 * A064610 A064611 A064612

Keyword

nonn,changed

Author

Labos Elemer, Sep 24 2001

Status

approved