A182394 Signs of differences of number of divisors function: a(n) = sign(d(n)-d(n-1)), cf. A000005.

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

Offset

2

Comments

d(n) (A000005) has offset 1, being an arithmetic function, so this sequence has offset 2.

Erdős proves that a(n) = 1 with natural density 1/2 and a(n) = -1 with natural density 1/2. Heath-Brown proved that a(n) = 0 infinitely often; see A005237 for details. - Charles R Greathouse IV, Oct 20 2013

Links

N. J. A. Sloane, Table of n, a(n) for n = 2..20000

P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530-540.

D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), pp. 141-149.

Formula

a(n) = 1 if d(n) > d(n - 1) and a(n) = -1 if d(n) < d(n - 1), otherwise a(n) = 0 if d(n) = d(n - 1), where d(n) is the number of divisors of n (A000005).

Example

The initial values d(1) ... d(20) are
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, ...
and the first differences are
1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, ...,
the signs of which are +1, 0, +1, -1, ...

Mathematica

Sign[Differences[DivisorSigma[0, Range[2..100]]]] (* T. D. Noe, Apr 27 2012, amended by N. J. A. Sloane, Oct 05 2017 *)

Prog

(PARI) a(n)=sign(numdiv(n)-numdiv(n-1)) \\ Charles R Greathouse IV, Oct 20 2013

Crossrefs

Cf. A000005, A051950, A175150 (accumulated sums).

Sequence in context: A128973 A176412 A013596 * A079054 A131695 A324113

Adjacent sequences: A182391 A182392 A182393 * A182395 A182396 A182397

Keyword

sign,easy

Author

Giovanni Teofilatto, Apr 27 2012

Extensions

Edited by N. J. A. Sloane, Oct 05 2017

Status

approved