A332682 -id:A332682 - cp's OEIS frontend

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

Ilya Gutkovskiy, May 25 2020

nonn

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

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

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

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

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

a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020

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

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.

A333505 a(n) = Sum_{k=1..n} (-1)^(k+1) * k * ceiling(n/k).

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 5, 5, 1, 4, 9, 9, 2, 2, 9, 17, 5, 5, 11, 11, 2, 12, 23, 23, -4, 1, 14, 26, 15, 15, 22, 22, -6, 8, 25, 37, 9, 9, 28, 44, 7, 7, 18, 18, 3, 35, 58, 58, -9, -2, 18, 38, 21, 21, 36, 52, 5, 27, 56, 56, -3, -3, 28, 68, 8, 26, 45, 45, 24, 50, 73, 73, -23, -23, 14

Offset: 1

Ilya Gutkovskiy, May 25 2020

sign

Cf. A001057, A002129, A006590, A024916, A024919, A332490, A332682.

Table[Sum[(-1)^(k + 1) k Ceiling[n/k], {k, 1, n}], {n, 1, 75}]
Table[(-1)^(n + 1) Ceiling[n/2] + Sum[DivisorSum[k, (-1)^(# + 1) # &], {k, 1, n - 1}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[x/(1 - x) (1/(1 + x)^2 + Sum[(-1)^(k + 1) k x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (-1)^(k+1)*k*ceil(n/k)); \\ Michel Marcus, May 26 2020

from math import isqrt
def A333505(n): return ((s:=isqrt(m:=n-1>>1))**2*(s+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))<<1)-((t:=isqrt(n-1))**2*(t+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,t+1))>>1) + (m+1 if n&1 else -m-1) # Chai Wah Wu, Oct 30 2023

G.f.: (x/(1 - x)) * (1/(1 + x)^2 + Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
a(n) = (-1)^(n+1) * ceiling(n/2) + Sum_{k=1..n-1} A002129(k).
a(n) = A001057(n) - A024919(n-1).

A332683 a(n) = Sum_{k=1..n, gcd(n, k) = 1} ceiling(n/k).

Original entry on oeis.org

1, 2, 5, 6, 12, 8, 20, 15, 23, 18, 37, 19, 47, 28, 38, 37, 66, 31, 76, 41, 61, 52, 96, 44, 96, 63, 89, 66, 129, 49, 141, 84, 109, 88, 129, 72, 176, 101, 132, 95, 198, 77, 210, 116, 142, 129, 232, 99, 226, 122, 186, 144, 269, 114, 232, 149, 214, 169, 305, 110

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Moebius transform of A006590.

Cf. A006590, A008683, A116477, A332682.

Table[Sum[Boole[GCD[n, k] == 1] Ceiling[n/k], {k, 1, n}], {n, 1, 60}]

a(n) = sum(k=1, n, if (gcd(n, k) == 1, ceil(n/k))); \\ Michel Marcus, Feb 21 2020

a(n) = Sum_{d|n} mu(n/d) * A006590(d).

cp's OEIS Frontend

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

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A333505 a(n) = Sum_{k=1..n} (-1)^(k+1) * k * ceiling(n/k).

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Python

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A332683 a(n) = Sum_{k=1..n, gcd(n, k) = 1} ceiling(n/k).

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