A164510 -id:A164510 - cp's OEIS frontend

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

Shyam Sunder Gupta, Jun 12 2002

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

			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).

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.

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

Cf. A164510.

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

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

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 *)

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

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

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

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

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

A306863 a(n) is the number of primes between the n-th and (n+1)-st odd composite numbers.

Original entry on oeis.org

2, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

Zhandos Mambetaliyev, Mar 14 2019

nonn

			The first few odd composite numbers are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49. Between 9 and 15, there are two primes (11 and 13); between 15 and 21 there are also two primes (17 and 19); between 21 and 25 there is only one prime (23), etc.

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

Cf. A071904, A000040, A001223 (differences between primes), A164510, A001097.

Differences@ PrimePi@ Complement[Range[3, #, 2], Prime@ Range[2, PrimePi@ #]] &@ 300 (* Michael De Vlieger, Apr 21 2019 *)
Differences[PrimePi/@Select[Range[3,301,2],CompositeQ]] (* Harvey P. Dale, Sep 16 2023 *)

{ b=9; for(i=6, 150, if(isprime(2*i-1)==0, print1(primepi(2*i-1)-primepi(b), ", "); b=2*i-1)) }

from itertools import count
from sympy import primepi, isprime
def A306863(n):
    if n == 1: return 2
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return next(c for c in count(0) if not isprime(m+(c<<1)+2)) # Chai Wah Wu, Aug 02 2024

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A071904 Odd composite numbers.

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A306863 a(n) is the number of primes between the n-th and (n+1)-st odd composite numbers.

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Mathematica

PARI

Python