A332682 a(n) = Sum_{k=1..n} (-1)^(k+1) * ceiling(n/k).

1, 1, 2, 3, 3, 4, 5, 6, 5, 7, 8, 9, 8, 9, 10, 13, 11, 12, 13, 14, 13, 16, 17, 18, 15, 17, 18, 21, 20, 21, 22, 23, 20, 23, 24, 27, 25, 26, 27, 30, 27, 28, 29, 30, 29, 34, 35, 36, 31, 33, 34, 37, 36, 37, 38, 41, 38, 41, 42, 43, 40, 41, 42, 47, 43, 46, 47, 48, 47, 50

Offset: 1

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006590, A048272, A059851, A275495, A325937.

Programs

Maple

f:= proc(n) local k; add((-1)^(k+1)*ceil(n/k),k=1..n) end proc:
map(f, [$1..100]); # Robert Israel, Nov 25 2024

Mathematica

Table[Sum[(-1)^(k + 1) Ceiling[n/k], {k, 1, n}], {n, 1, 70}]
nmax = 70; CoefficientList[Series[(x/(1 - x)) (1 + Sum[x^k/(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x] // Rest

PARI

a(n) = sum(k=1, n, (-1)^(k+1)*ceil(n/k)); \\ Michel Marcus, Feb 21 2020

Formula

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

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

a(n) = (n mod 2) + Sum_{k=1..n-1} A048272(k).

a(n) = 1 + Sum_{k<=n-1} A325937(k). - Robert Israel, Nov 25 2024