A120486 Partial sums of A000188.

1, 2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 17, 18, 19, 20, 24, 25, 28, 29, 31, 32, 33, 34, 36, 41, 42, 45, 47, 48, 49, 50, 54, 55, 56, 57, 63, 64, 65, 66, 68, 69, 70, 71, 73, 76, 77, 78, 82, 89, 94, 95, 97, 98, 101, 102, 104, 105, 106, 107, 109, 110, 111, 114, 122, 123, 124, 125, 127, 128

Offset

1,2

Comments

This sequence can also be described as the number of 3-term nondecreasing geometric progressions with no term exceeding n.

a(n) = A132188(n) - A132345(n). - Reinhard Zumkeller, Apr 21 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio

Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 35 (3) 2008, p 189-194.

Formula

a(n) = 3n log(n) / Pi^2 + O(n). - Griffin N. Macris, Jan 28 2017

a(n) ~ 3*n*((log(n) + (3*gamma - 1))/ Pi^2 - 12*(Zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019

a(n) = Sum_{k=1..floor(sqrt(n))} phi(k)*floor(n/k^2), where phi is the Euler totient function A000010. - Ridouane Oudra, Aug 18 2019

G.f.: (1/(1 - x)) * Sum_{k>=1} phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 26 2021

From Ridouane Oudra, Oct 05 2024: (Start)

a(n) = Sum_{i=1..n} Sum_{j=1..i} A010052(i*j).

a(n) = A132345(n) + n.

a(n) = (1/2)*A132189(n) + n.

a(n) = (1/2)*(A132188(n) + n). (End)

Maple

with(numtheory): seq(add(phi(k)*floor(n/k^2), k=1..floor(sqrt(n))), n=1..100); # Ridouane Oudra, Aug 18 2019

Prog

(Haskell)
a120486 n = a120486_list !! (n - 1)
a120486_list = scanl1 (+) a000188_list
-- Reinhard Zumkeller, Apr 22 2012
(Python)
# uses Python function from A002088
from math import isqrt
def A120486(n):
    c, j, v = 0, 1, 0
    while (j2:=j**2) <= n:
        k = n//j2
        m = isqrt(n//k)
        c += k*(-v+(v:=A002088(m)))
        j = m+1
    return c # Chai Wah Wu, May 16 2026

Crossrefs

Cf. A000188, A132188, A132189, A132345, A010052.

Sequence in context: A028748 A028783 A274779 * A229993 A376592 A323252

Adjacent sequences: A120483 A120484 A120485 * A120487 A120488 A120489

Keyword

nonn

Author

Gerry Myerson, Nov 21 2007

Status

approved