A120885 -id:A120885 - 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.

A138618 Triangle of exponentials of Mangoldt function M(n) read by rows, in which row products give the natural numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 5, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik, May 14 2008

Row sums are A001414. This table is similar to A139547 and A120885.

Cumulative column products are A003418, A139550, A139552, A139554.

			1 = 1
2*1 = 2
3*1*1 = 3
2*2*1*1 = 4
5*1*1*1*1 = 5
1*3*2*1*1*1 = 6
7*1*1*1*1*1*1 = 7
2*2*1*2*1*1*1*1 = 8
3*1*3*1*1*1*1*1*1 = 9
1*5*1*1*2*1*1*1*1*1 = 10
11*1*1*1*1*1*1*1*1*1*1 = 11
1*1*2*3*1*2*1*1*1*1*1*1 = 12
13*1*1*1*1*1*1*1*1*1*1*1*1 = 13

Eric Weisstein's World of Mathematics, Mangoldt Function.

Cf. A120885, A139547, A001414, A000027.

Flatten[Table[Table[If[Mod[n, k] == 0, Exp[MangoldtLambda[n/k]], 1], {k, 1, n}], {n, 1, 14}]] (* Mats Granvik, May 23 2013 *)

M(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
T(n,k) = if (n % k, 1, M(n/k));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Mar 03 2023

T(n,k) = A014963(n/k) if n mod k = 0, otherwise 1. - Mats Granvik, May 23 2013

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 11, 11, 19, 16, 21, 21, 30, 30, 37, 29, 45, 45, 51, 51, 66, 56, 67, 67, 88, 83, 96, 84, 105, 105, 112, 112, 144, 130, 147, 135, 159, 159, 178, 162, 197, 197, 208, 208, 241, 209, 232, 232, 277, 270, 290, 270, 309, 309, 324, 308, 357, 335, 364, 364

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

Cf. A000217, A000593, A078471, A120885, A330926, A332490.

b:= n-> add(d, d=select(x-> x::odd, numtheory[divisors](n))):
a:= proc(n) option remember; n+`if`(n<2, 0, a(n-1))-b(n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2] k, {k, 1, n}], {n, 1, 60}]
Table[n (n + 1)/2 - Sum[DivisorSum[k, (-1)^(k/# + 1) # &], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x) (1/(1 - x)^2 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (ceil(n/k) % 2)*k); \\ Michel Marcus, May 26 2020

G.f.: (x/(1 - x)) * (1/(1 - x)^2 - Sum_{k>=1} k * x^k / (1 + x^k)).
a(n) = n*(n + 1)/2 - Sum_{k=1..n-1} A000593(k).
a(n) = A000217(n) - A078471(n-1).

cp's OEIS Frontend

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

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A138618 Triangle of exponentials of Mangoldt function M(n) read by rows, in which row products give the natural numbers.

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Mathematica

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A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

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Maple

Mathematica

PARI

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