

A002808


The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
(Formerly M3272 N1322)


917



4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88
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OFFSET

1,1


COMMENTS

The natural numbers 1,2,... are divided into three sets: 1 (the unit), the primes (A000040) and the composite numbers (A002808).
The number of composite numbers <= n (A065855) = n  pi(n) (A000720)  1.
n is composite iff sigma(n) + phi(n) > 2n. This is a nice result of the well known theorem: For all positive integers n, n = Sum_{dn} phi(d). For the proof see my contribution to puzzle 76 of Carlos Rivera's Primepuzzles.  Farideh Firoozbakht, Jan 27 2005, Jan 18 2015
The composite numbers have the semiprimes A001358 as primitive elements.
Composite numbers n which are the product of r=A001222(n) prime numbers are sometimes called ralmost primes. Sequences listing ralmost primes are: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).  Jason Kimberley, Oct 02 2011
Degrees for which there are irreducible polynomials which are reducible mod p for all primes p, see Brandl.  Charles R Greathouse IV, Sep 04 2014
An integer is composite if and only if it is the sum of strictly positive integers in arithmetic progression with common difference 2: 4 = 1 + 3, 6 = 2 + 4, 8 = 3 + 5, 9 = 1 + 3 + 5, etc.  JeanChristophe Hervé, Oct 02 2014
This statement holds since k+(k+2)+...+k+2(n1) = n*(n+k1) = a*b with arbitrary a,b (taking n=a and k=ba+1 if b>=a).  M. F. Hasler, Oct 04 2014
For n > 4, these are numbers n such that n!/n^2 = (n1)!/n is an integer (see A056653).  Derek Orr, Apr 16 2015
Let f(x) = Sum_{i=1..x} Sum_{j=2..i1} cos((2*Pi*x*j)/i). It is known that the zeros of f(x) are the prime numbers. So these are the numbers n such that f(n) > 0.  Michel Lagneau, Oct 13 2015
Numbers n that can be written as solutions of the Diophantine equation n = (x+2)(y+2) where {x,y} in N^2, pairs of natural numbers including zero (cf. Mathematica code and Davis).  Ron R Spencer and Bradley Klee, Aug 15 2016
Numbers n with a partition (containing at least two summands) so that its summands also multiply to n. If n is prime, there is no way to find those two (or more) summands. If n is composite, simply take a factor or several, write those divisors and fill with enough 1's so that they add up to n. For example: 4 = 2*2 = 2+2, 6 = 1*2*3 = 1+2+3, 8 = 1*1*2*4 = 1+1+2+4, 9 = 1*1*1*3*3 = 1+1+1+3+3.  Juhani Heino, Aug 02 2017


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 2.
A. E. Bojarincev, Asymptotic expressions for the nth composite number, Univ. Mat. Zap. 6:2143 (1967).  In Russian.
Martin Davis, "Algorithms, Equations, and Logic", pp. 415 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 66.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 51.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

a(n) = pi(a(n)) + 1 + n, where pi is the prime counting function.
n + n/log n + n/log^2 n < a(n) < n + n/log n + 3n/log^2 n for n >= 4, see Panaitopol. Bojarincev gives an asymptotic version.  Charles R Greathouse IV, Oct 23 2012
a(n) = 1 + a(n1) + f(n) for n > 1 with a(1) = 4 where f(n) = 1 if n belongs to A014689, 0 otherwise.  Mikhail Kurkov, Dec 21 2021


MAPLE

t := []: for n from 2 to 20000 do if isprime(n) then else t := [op(t), n]; fi; od: t; remove(isprime, [$3..89]); # Zerinvary Lajos, Mar 19 2007
A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from procname(n1)+1 do if not isprime(a) then return a; end if; end do ; end if; end proc; # R. J. Mathar, Oct 27 2009


MATHEMATICA

Select[Range[2, 100], !PrimeQ[#]&] (* Zak Seidov, Mar 05 2011 *)
With[{nn=100}, Complement[Range[nn], Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, May 01 2012 *)


PROG

(PARI) A002808(n)= my(k=1); while( n + n += k + k=primepi(n), ); n \\ For n=10^4 resp. 3*10^4, this is about 100 resp. 500 times faster than the former; M. F. Hasler, Nov 11 2009
(PARI) forcomposite(n=1, 1e2, print1(n, ", ")) \\ Felix Fröhlich, Aug 03 2014
(PARI) for(n=1, 1e3, if(bigomega(n) > 1, print1(n, ", "))) \\ Altug Alkan, Oct 14 2015
(Haskell)
a002808 n = a002808_list !! (n1)
a002808_list = filter ((== 1) . a066247) [2..]
(Python)
from sympy import primepi
m, k = n, primepi(n) + 1 + n
while m != k:
m, k = k, primepi(k) + 1 + n
return m # Chai Wah Wu, Jul 15 2015, updated Apr 14 2016
(Python)
from sympy import isprime
def ok(n): return n > 1 and not isprime(n)


CROSSREFS



KEYWORD

nonn,nice,easy,core


AUTHOR



EXTENSIONS



STATUS

approved



