A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n).

1, 1, 2, 3, 6, 7, 10, 11, 14, 19, 20, 25, 28, 29, 32, 37, 42, 43, 48, 51, 52, 57, 60, 65, 72, 75, 76, 79, 80, 83, 96, 99, 104, 105, 114, 115, 120, 125, 128, 133, 138, 139, 148, 149, 152, 153, 164, 175, 178, 179, 182, 187, 188, 197, 202, 207, 212, 213, 218, 221, 222

Offset: 1

Author

Mohammad K. Azarian

Keywords

• nonn
• easy
• nice

Comments

a(n) = A048864(A000040(n)) = number of nonprimes in RRS of n-th prime. - Labos Elemer, Oct 10 2002

A000040 - A014689 = A000027; in other words, the sequence of natural numbers subtracted from the prime sequence produces A014689. - Enoch Haga, May 25 2009

a(n) = A000040(n) - n. a(n) = inverse (frequency distribution) sequence of A073425(n), i.e., number of terms of sequence A073425(n) less than n. a(n) = A065890(n) + 1, for n >= 1. a(n) - 1 = A065890(n) = the number of composite numbers, i.e., (A002808) less than n-th primes, (i.e., < A000040(n)). - Jaroslav Krizek, Jun 27 2009

a(n) = A162177(n+1) + 1, for n >= 1. a(n) - 1 = A162177(n+1) = the number of composite numbers, i.e., (A002808) less than (n+1)-th number of set {1, primes}, (i.e., < A008578(n+1)). - Jaroslav Krizek, Jun 28 2009

Conjecture: Each residue class contains infinitely many terms of this sequence. Similarly, for any integers m > 0 and r, we have prime(n) + n == r (mod m) for infinitely many positive integers n. - Zhi-Wei Sun, Nov 25 2013

First differences are A046933 = differences minus one between successive primes. - Gus Wiseman, Jan 18 2020

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Equals A014692 - 1.

Cf. A000040, A033286, A158611, A002808, A065890.

Cf. A232463, A232443.

The sum of prime factors of n is A001414(n).

The sum of prime indices of n is A056239(n).

Their difference is A331415(n).

Cf. A000720, A046933, A056239, A318995, A325036, A331380, A331416, A331418.

Programs

Haskell

a014689 n = a000040 n - fromIntegral n
-- Reinhard Zumkeller, Apr 09 2012

Magma

[NthPrime(n)-n: n in [1..70]]; // Vincenzo Librandi, Mar 20 2013

Mathematica

Table[Prime[n] - n, {n, 61}] (* Alonso del Arte *)

PARI

a(n) = prime(n)-n \\ Charles R Greathouse IV, Sep 05 2011

Python

from sympy import prime
def A014689(n): return prime(n)-n # Chai Wah Wu, Oct 11 2024

Formula

G.f: b(x) - x/((1-x)^2), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

Extensions

More terms from Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010