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

1,1

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

Links

Zak Seidov and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)

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

Example

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).

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 *)

Prog

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

Crossrefs

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

Cf. A164510.

Sequence in context: A160666 A039769 A270574 * A326586 A014076 A067800

Adjacent sequences: A071901 A071902 A071903 * A071905 A071906 A071907

Keyword

nice,nonn,easy

Author

Shyam Sunder Gupta, Jun 12 2002

Status

approved