A357385 a(n) = A071626(n+1) - A071626(n).

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

Offset

1

Comments

Conjectures:

1) The only distinct terms of this sequence are -1, 0 and 1.

2) The asymptotic density of the occurrences of 0 is 0.943..., and the asymptotic densities of the occurrences of -1 and 1 are equal and the common value is 0.028... .

These conjectures are empirically supported by the following numbers of occurrences of the terms -1, 0 and 1 up to n = 10^k:

k -1 0 1

- ------- ------- ------

1 0 6 4

2 1 87 12

3 28 913 59

4 362 9189 449

5 3159 93430 3411

6 27853 943571 28576

7 281802 9434309 283889

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(p) = 0 for an odd prime p.

Mathematica

s[n_] := Length[Union[FactorInteger[n!][[;; , 2]]]]; s[1] = 0; Differences[Table[s[n], {n, 1, 100}]]

Crossrefs

Cf. A071626.

Sequence in context: A029694 A171894 A284957 * A316533 A085405 A036988

Adjacent sequences: A357382 A357383 A357384 * A357386 A357387 A357388

Keyword

sign

Author

Amiram Eldar, Sep 26 2022

Status

approved