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

Author

Gus Wiseman, Jul 06 2019

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

Magma

[1] cat [1+(&+[(-1)^(k)*#Divisors(k):k in [1..n]]):n in [1..70]]; // Marius A. Burtea, Jul 10 2019

Mathematica

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

PARI

a(n) = 1 - sum(k=1, n, (-1)^(k+1)*numdiv(k)); \\ Michel Marcus, Jul 09 2019

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