A073169 -id:A073169 - cp's OEIS frontend

A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

0, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n-1) = A018252(n) - n. a(n-1) = inverse (frequency distribution) sequence of A014689(n), i.e. number of terms of sequence A014689(n) less than n. a(n) = A073169(n+1) - 1, for n >= 1. For n >= 1: a(n) + 1 = A073169(n) = the number of set {1, primes}, i.e. (A008578) less than (n)-th composite numbers (A002828(n)). a(n-1) = The number of primes (A000040(n)) less than n-th nonprime (A018252(n)). - Jaroslav Krizek, Jun 27 2009

			n=100: composite[100]=133,Pi[133]=32=a(100)

Cf. A065890, A073426, A000720, A002808, A000040, A018252, A158611, A073169.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[PrimePi[c[w]], {w, 1, 128}]
With[{nn=150},PrimePi/@Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, Jun 26 2013 *)

from sympy import composite
def A073425(n): return composite(n)-n-1 if n else 0 # Chai Wah Wu, Oct 11 2024

a(n) = A000720(A002808(n)).
a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015
a(n) = A002808(n)-n-1 for n > 0. - Chai Wah Wu, Oct 11 2024

Edited by N. J. A. Sloane, Jul 04 2009 at the suggestion of R. J. Mathar

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

A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 9, 10, 13, 18, 19, 24, 27, 28, 31, 36, 41, 42, 47, 50, 51, 56, 59, 64, 71, 74, 75, 78, 79, 82, 95, 98, 103, 104, 113, 114, 119, 124, 127, 132, 137, 138, 147, 148, 151, 152, 163, 174, 177, 178, 181, 186, 187, 196, 201, 206, 211, 212, 217, 220, 221

Offset: 1

Jaroslav Krizek, Jun 27 2009

Essentially the same as A065890.

a(n) = number of terms of A073169(n) less than n.

			A008578(6) = 11, and there are 5 composites smaller than 11, viz. 4, 6, 8, 9, 10, hence a(6) = 5.

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

Cf. A002808 (composites), A008578 (1 and the primes), A065890, A073169.

T:=[0,1] cat PrimesUpTo(300); [ T[n+1]-n: n in [1..#T-1] ]; // Klaus Brockhaus, Sep 08 2009

Join[{0},Module[{nn=300,cmps},cmps=Accumulate[Table[If[CompositeQ[n],1,0],{n,nn}]];Table[cmps[[p]],{p,Prime[ Range[ PrimePi[ nn]]]}]]] (* Harvey P. Dale, Nov 11 2024 *)

a(n) = A008578(n) - n = A158611(n+1) -n.

a(n) = A065890(n-1) for n > 1.

Edited and extended by Klaus Brockhaus, Sep 09 2009

A073170 a(1) = a(2) = 0; for n>2, a(n) = prime(n-1)-n+1.

Original entry on oeis.org

0, 0, 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

Offset: 1

Labos Elemer, Jul 19 2002

nonn

Smallest x such that A002808(x)-x (cf. A073169) equals n, or 0 if no such value exists.

Cf. A002808, A073168, A073169. Essentially same sequence as A014689.

Cf. A000040, A014689.

f[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] t=Table[0, {100}]; Do[s=f[n]-n; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10000}]; t
Join[{0,0},Table[Prime[n-1]-n+1,{n,3,70}]] (* Harvey P. Dale, Dec 18 2012 *)

a(n)=Min{x; composite[x]-x=n}

Simpler description from Vladeta Jovovic, Mar 29 2003

Definition corrected by Harvey P. Dale, Dec 18 2012

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

A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A073170 a(1) = a(2) = 0; for n>2, a(n) = prime(n-1)-n+1.

Views

Author

Keywords

Comments

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