A162177 -id:A162177 - cp's OEIS frontend

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

Mohammad K. Azarian

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

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

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.

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

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

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

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

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

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

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

A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

Original entry on oeis.org

3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 26, 26

Offset: 1

Labos Elemer, Jul 19 2002

nonn

a(n) = the number of numbers of set {1, prime} (A008578(n)) less than n-th composite numbers (A002808(n)). a(n) = inverse (frequency distribution) sequence of A162177(n), i.e. number of terms of sequence A162177(n) less than n for n >= 1. a(n) = A002808(n) + A162177(n) - A158611(n+1) for n >= 1. a(n) = A002808(n) + A162177(n) - A008578(n) for n >= 1. [From Jaroslav Krizek, Jul 23 2009]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007821, A022449, A000040, A002808, A073168.

c[n_Integer] := FixedPoint[n+PrimePi[ # ]+1&, n] Table[c[w]-w, {w, 1, 128}]
With[{c=Select[Range[100],CompositeQ]},#[[1]]-#[[2]]&/@Thread[ {c,Range[ Length[ c]]}]] (* Harvey P. Dale, Feb 03 2015 *)

a(n)=1+A073425(n). [From R. J. Mathar, Jul 31 2009]

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

A163248 Sum of the n-th composite number plus the number of composite numbers less than the n-th noncomposite number.

Original entry on oeis.org

4, 6, 8, 10, 12, 17, 20, 24, 26, 31, 38, 40, 46, 51, 53, 57, 63, 69, 72, 79, 83, 85, 91, 95, 102, 110, 114, 117, 122, 124, 128, 143, 147, 153, 155, 165, 168, 174, 180, 184, 190, 197, 200, 210, 212, 216, 218, 231, 243, 247

Offset: 1

Jaroslav Krizek, Jul 23 2009

Bo Gyu Jeong, Table of n, a(n) for n = 1..10000

nnc:= 1: nc:= 0; b[nnc]:= 0:b[0]:= 0:
for x from 2 to 1000 do
   if isprime(x) then
     nnc:= nnc+1; b[nnc]:= b[nnc-1];
   else
     b[nnc]:= b[nnc]+1;
     nc:= nc+1;
     c[nc]:= x;
   fi
od:
seq(b[n-1]+c[n],n=1..min(nnc,nc)); # Robert Israel, Jan 09 2015

a(n) = A002808(n) + A162177(n) = A008578(n) + A073169(n).

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Formula

Extensions

A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A163248 Sum of the n-th composite number plus the number of composite numbers less than the n-th noncomposite number.

Views

Author

Keywords

Links

Programs

Maple

Formula

Extensions