A309081 -id:A309081 - cp's OEIS frontend

A309082 a(n) = n - floor(n/2^3) + floor(n/3^3) - floor(n/4^3) + ...

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67

Offset: 1

Ilya Gutkovskiy, Jul 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A013937, A059851, A061704, A309081, A309083.

Table[Sum[(-1)^(k + 1) Floor[n/k^3], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
Table[Sum[Boole[IntegerQ[d^(1/3)] && OddQ[d]], {d, Divisors[n]}] - Sum[Boole[IntegerQ[d^(1/3)] && EvenQ[d]], {d, Divisors[n]}], {n, 1, 75}] // Accumulate

G.f.: (1/(1 - x)) * Sum_{k>=1} (-1)^(k+1) * x^(k^3)/(1 - x^(k^3)).

a(n) ~ 3*zeta(3)*n/4. - Vaclav Kotesovec, Oct 12 2019

A309083 a(n) = n - floor(n/2^4) + floor(n/3^4) - floor(n/4^4) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Ilya Gutkovskiy, Jul 11 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A013938, A059851, A063775, A309081, A309082.

Table[Sum[(-1)^(k + 1) Floor[n/k^4], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
Table[Sum[Boole[IntegerQ[d^(1/4)] && OddQ[d]], {d, Divisors[n]}] - Sum[Boole[IntegerQ[d^(1/4)] && EvenQ[d]], {d, Divisors[n]}], {n, 1, 75}] // Accumulate

G.f.: (1/(1 - x)) * Sum_{k>=1} (-1)^(k+1) * x^(k^4)/(1 - x^(k^4)).

a(n) ~ 7*zeta(4)*n/8 = 7*Pi^4*n/720. - Vaclav Kotesovec, Oct 12 2019

A350221 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^2).

Original entry on oeis.org

1, 3, 8, 12, 21, 29, 40, 52, 67, 83, 100, 116, 140, 160, 185, 210, 237, 264, 298, 327, 363, 397, 435, 472, 514, 557, 602, 644, 690, 741, 791, 837, 897, 950, 1009, 1063, 1126, 1185, 1253, 1313, 1381, 1450, 1521, 1593, 1667, 1739, 1820, 1894, 1973, 2054, 2140

Offset: 1

Seiichi Manyama, Dec 20 2021

nonn

Cf. A000290, A153818, A309081, A344720, A350222, A350224.

a[n_] := Sum[(-1)^(k + 1)*Floor[(n/k)^2], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 20 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n^2\k^2));

def A350221(n): return (m:=n**2)+sum(m//k**2 if k&1 else -(m//k**2) for k in range(2,n+1)) # Chai Wah Wu, Oct 27 2023

a(n) = A309081(n^2).

cp's OEIS Frontend

A309082 a(n) = n - floor(n/2^3) + floor(n/3^3) - floor(n/4^3) + ...

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A309083 a(n) = n - floor(n/2^4) + floor(n/3^4) - floor(n/4^4) + ...

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A350221 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^2).

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