A326440 a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.

1, 0, 2, 0, 3, 1, 5, 3, 7, 4, 8, 6, 12, 10, 14, 10, 15, 13, 19, 17, 23, 19, 23, 21, 29, 26, 30, 26, 32, 30, 38, 36, 42, 38, 42, 38, 47, 45, 49, 45, 53, 51, 59, 57, 63, 57, 61, 59, 69, 66, 72, 68, 74, 72, 80, 76, 84, 80, 84, 82, 94, 92, 96, 90, 97, 93, 101, 99

Offset

0,3

Comments

Is this sequence nonnegative?

As tau(n) is odd when n is a square, there are alternating strings of even and odd integers with change of parity for each n square. Indeed, between m^2 and (m+1)^2-1, there is a string of 2m+1 even terms if m is odd, or a string of 2m+1 odd terms if m is even. - Bernard Schott, Jul 10 2019

Links

Michel Marcus, Table of n, a(n) for n = 0..5000

Formula

a(n) = 1 + Sum_{k=1..n} (-1)^k A000005(k).

For n > 0, a(n) = 1 + A307704(n).

If p prime, a(p) = a(p-1) - 2. - Bernard Schott, Jul 10 2019

Example

The first 6 terms of A000005 are 1, 2, 2, 3, 2, 4, so a(6) = 1 - 1 + 2 - 2 + 3 - 2 + 4 = 5.

Mathematica

Accumulate[Table[If[k==0, 1, (-1)^k*DivisorSigma[0, k]], {k, 0, 30}]]

Prog

(PARI) a(n) = 1 - sum(k=1, n, (-1)^(k+1)*numdiv(k)); \\ Michel Marcus, Jul 09 2019
(Magma) [1] cat [1+(&+[(-1)^(k)*#Divisors(k):k in [1..n]]):n in [1..70]]; // Marius A. Burtea, Jul 10 2019

Crossrefs

Cf. A000005, A001222, A008683, A054519, A071321, A195017, A268387, A307704, A316523, A316524, A319273.

Sequence in context: A282892 A008798 A005290 * A166117 A078051 A130627

Adjacent sequences: A326437 A326438 A326439 * A326441 A326442 A326443

Keyword

nonn

Author

Gus Wiseman, Jul 06 2019

Status

approved