A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22

Offset

1,3

Comments

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).

a(n) = n - Sum_{k=1..n-1} A048272(k).

a(n) = A075997(n-1) + 1.

Maple

b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020

Mathematica

Table[Sum[Mod[Ceiling[n/k], 2], {k, 1, n}], {n, 1, 80}]
Table[n - Sum[DivisorSum[k, (-1)^(# + 1) &], {k, 1, n - 1}], {n, 1, 80}]
nmax = 80; CoefficientList[Series[x/(1 - x) (1 + Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020
(Python)
from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//k for k in range(1, s+1))-(sum(m//k for k in range(1, t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

Crossrefs

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

Sequence in context: A078709 A023022 A177501 * A323273 A100677 A188637

Adjacent sequences: A330923 A330924 A330925 * A330927 A330928 A330929

Keyword

nonn

Author

Ilya Gutkovskiy, May 25 2020

Status

approved