A381042 Alternating sum of floor(n^(1/k)), with k >= 2.

0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7

Offset: 0

Author

Friedjof Tellkamp, Apr 14 2025

Keywords

• nonn

Examples

			n:       0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
k=2 (+): 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, ... (A000196)
k=3 (-): 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, ... (A048766)
...
Sum:     0, 0, 0, 0, 1, 1, 1, 1, 0, 1 ... (= this sequence).
		

Links

• Friedjof Tellkamp, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000196 (k=2), A048766 (k=3), A255270 (k=4), A178487 (k=5), A178489 (k=6).

Cf. A089361 (nonalternating), A382691, A382692.

Programs

Mathematica

z = 100; Table[Sum[(-1)^k Floor[n^(1/k)], {k, 2, 2 Floor@Log[2, z/2] - 1}], {n, 0, z}]

Formula

a(n) = A000196(n) - A048766(n) + A255270(n) - A178487(n) + ... .

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

a(n) = Sum_{i=1..n} A382691(i).

a(n) ~ A382692(n).

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