A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4

Offset: 1

Author

Jaroslav Krizek, Sep 29 2010

Keywords

• nonn
• look

Comments

Sequence is cardinal and not fractal. Cardinal sequence is sequence with infinitely many times occurring all natural numbers. Fractal sequence is sequence such that when the first instance of each number in the sequence is erased, the original sequence remains.

Ordinal transform of the nextprime function, A151800(1..) = 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, ..., also ordinal transform of A304106. - Antti Karttunen, Jun 09 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000720, A007917, A008578, A151799, A151800, A305300.

Cf. A065358 for another way of visualizing prime gaps.

Cf. A304106 (ordinal transform of this sequence).

Cf. A049711.

Programs

Mathematica

a[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

PARI

A175851(n) = if(1==n,n,1 + n - precprime(n)); \\ Antti Karttunen, Mar 04 2018

Formula

a(1) = 1, a(n) = n - A007917(n) + 1 for n >= 2. a(1) = 1, a(2) = 1, a(n) = n - A151799(n+1) + 1 for n >= 3.

a(n) = Sum_{i=1..n} floor(pi(i)/pi(n)), for n>1 with pi(n) = A000720(n). - Ridouane Oudra, Jun 24 2024

a(n) = A049711(n+1), for n>1. - Ridouane Oudra, Jul 16 2024