A281487 a(n+1) = -Sum_{d|n} a(d), a(1) = 1.

1, -1, 0, -1, 1, -2, 2, -3, 4, -5, 4, -5, 8, -9, 7, -9, 13, -14, 12, -13, 18, -21, 17, -18, 29, -31, 23, -28, 36, -37, 36, -37, 50, -55, 42, -46, 64, -65, 53, -62, 83, -84, 75, -76, 94, -107, 90, -91, 129, -132, 107, -121, 145, -146, 135, -141, 180, -193, 157

Offset: 1

Author

Andrey Zabolotskiy, Jan 22 2017

Keywords

• sign
• easy

Comments

a(1) = 1, any other choice simply adds a factor to all terms.

Observations: sign of a(n) is -(-1)^n, the subsequences |a(n)| with n = 1, 2 mod 4 and |a(n)| with n = 3, 0 mod 4 both grow at n>5. Both these subsequences seem to share the asymptotics with A003238 (and hence A000123): log(|a(n)|) is approximately proportional to (log(n/log(n)))^2; however, the factor is much less than log(4).

There is a family of sequences with the formula a(n) = s*Sum_{d|(n-k), 1<=dA002033, A003238, A007439. For s=-1 and k = 0,1,2, these are the Möbius function A008683, this sequence, and A281488.

Examples

			a(9) = -(a(1)+a(2)+a(4)+a(8)) = -(1-1-1-3) = 4.
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A003238, A281488, A000123.

Programs

PARI

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n-1, d, va[d]);); va;} \\ Michel Marcus, Apr 29 2019

Python

a = [1]
for n in range(1, 100):
   a.append(-sum(a[d-1] for d in range(1, n+1) if n%d == 0))
print(a)

Formula

a(1) = 1.

a(n+1) = -Sum_{d|n} a(d) for n>=1.

a(n+1) = Sum_{d|n} |a(d)|*(-1)^(d+n) for n>=1.

From Ilya Gutkovskiy, Apr 29 2019: (Start)

G.f.: x * (1 - Sum_{n>=1} a(n)*x^n/(1 - x^n)).

L.g.f.: log(Product_{n>=1} (1 - x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n. (End)