A382692 -id:A382692 - cp's OEIS frontend

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

Friedjof Tellkamp, Apr 14 2025

nonn

			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).

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

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

Cf. A089361 (nonalternating), A382691, A382692.

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

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).

A382691 Alternating sum of the characteristic functions of k-th powers, with k >= 2: characteristic function of squares - c.f. of cubes + c.f. of 4th powers - ... .

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

Friedjof Tellkamp, Apr 05 2025

sign

			n:           1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Squares (+): 1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes   (-): 1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
...
Sum:         0, 0, 0, 1, 0, 0, 0,-1, 1, ... (= this sequence).

Friedjof Tellkamp, Table of n, a(n) for n = 1..10000
Solomon W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223.

Cf. A010052, A010057, A374016, A001597, A381042, A382692.

Cf. A089723 (nonalternating, k>=1), A259362 (nonalternating, k>=2).

Table[Sum[(-1)^k Boole[IntegerQ[n^(1/k)]], {k, 2, Floor[Log[2, n]]}], {n, 1, 100}]

a(n) = sum(i=2, logint(n,2), (-1)^i*ispower(n, i)); \\ Michel Marcus, Apr 11 2025

a(n) = A010052(n) - A010057(n) + A374016(n) - (...).
Sum_{i=1..n} a(i) = A381042(n).
G.f.: Sum_{j>=1, k>=2} (-1)^k * x^(j^k).
Sum_{n>=1} a(n)/n = 1/2.
Dirichlet g.f.: Sum_{k>=2} (-1)^k * zeta(k*s) = Sum_{k>=1} (zeta(2*k*s) - zeta((2*k+1)*s)).

cp's OEIS Frontend

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

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