A071904 Odd composite numbers.

9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205

Offset: 1

Author

Shyam Sunder Gupta, Jun 12 2002

Keywords

• nice
• nonn
• easy
• changed

Comments

Same as A014076 except for the initial term A014076(1)=1 (which is not a composite number).

Values of quadratic form (2x + 3)*(2y + 3) = 4xy + 6x + 6y + 9 for x, y >= 0. - Anton Joha, Jan 21 2001

Intersection of A002808 and A005408. - Reinhard Zumkeller, Oct 10 2011

Composite numbers n such that (n-1)^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012

There is a rectangular array of n dots (with both sides > 1) with a unique center point if and only if n is in this sequence. - Peter Woodward, Apr 21 2015

First differences <= 6. Cf. A164510. - Zak Seidov, Sep 22 2016

Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (4/5) * (8/7) * (10/11) * (12/13) * (14/13) * ... = Pi/4. - Dimitris Valianatos, May 24 2017

Examples

			45 is in the sequence because it is odd and composite (45 = 3 * 3 * 5).
195 is in the sequence because it is odd and composite (195 = 3 * 5 * 13).
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov).
• Joel E. Cohen and Dexter Senft, Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often, Notes on Number Theory and Discrete Mathematics, Volume 31, Number 3, 494-503 (2025). See p. 495.

Crossrefs

Cf. A002808, A014076, A005408, A000035, A010051, A020639, A162022.

Cf. A164510.

Programs

Haskell

a071904 n = a071904_list !! (n-1)
a071904_list = filter odd a002808_list
-- Reinhard Zumkeller, Oct 10 2011

Maple

remove(isprime, [seq(2*i+1, i = 1 .. 1000)]); # Robert Israel, Apr 22 2015
# alternative
A071904 := proc(n) local a;
    if n = 1 then
        9;
    else
        for a from procname(n-1)+2 by 2 do
            if not isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 09 2015

Mathematica

Select[Table[n, {n, 9, 300, 2}], !PrimeQ[#] &] (* Vladimir Joseph Stephan Orlovsky, Apr 16 2011 *)
With[{upto = 200}, Complement[Range[9, upto, 2], Prime[Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Jan 24 2013 *)
With[{upto = 200},oddsequence=Table[2n+1,{n,1,upto}];oddcomposites=Union[Flatten[Range[oddsequence^2,upto,2*oddsequence]]]] (* Ben Engelen, Feb 24 2016 *)

PARI

is(n)=n%2 && !isprime(n) && n > 1 \\ Charles R Greathouse IV, Nov 24 2012

PARI

lista(nn) = forcomposite(n=1, nn, if (n%2, print1(n, ", "))); \\ Michel Marcus, Sep 24 2016

Python

from sympy import isprime
def ok(n): return n > 3 and n%2 == 1 and not isprime(n)
print(list(filter(ok, range(206)))) # Michael S. Branicky, Sep 15 2021

Python

from sympy import primepi
def A071904(n):
    if n == 1: return 9
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m # Chai Wah Wu, Jul 31 2024

Formula

A000035(a(n))*(1-A010051(a(n))) = 1; A020639(a(n)) = A162022(n). - Reinhard Zumkeller, Oct 10 2011

a(n) ~ 2n. - Charles R Greathouse IV, Jul 02 2013

More precisely, a(n) = 2n(1 + 2(1+o(1))/log(n)). - Vladimir Shevelev, Jan 07 2015