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A014076 Odd nonprimes.

%I A014076 #94 Aug 04 2024 05:34:36
%S A014076 1,9,15,21,25,27,33,35,39,45,49,51,55,57,63,65,69,75,77,81,85,87,91,
%T A014076 93,95,99,105,111,115,117,119,121,123,125,129,133,135,141,143,145,147,
%U A014076 153,155,159,161,165,169,171,175,177,183,185,187,189,195,201,203,205,207
%N A014076 Odd nonprimes.
%C A014076 Same as A071904 except for the initial term 1 (which is not composite).
%C A014076 Numbers n such that product of first n odd numbers divided by sum of the first n odd numbers is an integer : 1*3*5*...*(2*n - 1) / (1 + 3 + 5 + ... + (2*n - 1)) = c. - _Ctibor O. Zizka_, Jun 26 2010
%C A014076 Conjecture: There exist infinitely many pairs [a(n), a(n)+6] such that a(n)/3 and (a(n)+6)/3 are twin primes. - _Eric Desbiaux_, Sep 25 2014.
%C A014076 Odd numbers 2*n + 1 such that (2*n)!/(2*n + 1) is an integer. Odd terms of A056653. - _Peter Bala_, Jan 24 2017
%H A014076 Reinhard Zumkeller, <a href="/A014076/b014076.txt">Table of n, a(n) for n = 1..10000</a>
%F A014076 A000035(a(n))*(1 - A010051(a(n))) = 1. - _Reinhard Zumkeller_, Sep 30 2011
%F A014076 a(n) ~ 2n. - _Charles R Greathouse IV_, Jul 02 2013
%F A014076 (a(n+2)-1)/2 - pi(a(n+2)-1) = n. - _Anthony Browne_, May 25 2016. Proof from Robert Israel: This follows by induction on n. If f(n) = (a(n+2)-1)/2 - pi(a(n+2)-1), one can show f(n+1) - f(n) = 1 (there are three cases to consider, depending on primeness of a(n+2) + 2 and a(n+2) + 4).
%F A014076 Union of A091113 and A091236. - _R. J. Mathar_, Oct 02 2018
%p A014076 remove(isprime, [seq(i,i=1..1000,2)]); # _Robert Israel_, May 25 2016
%p A014076 for n from 0 to 120 do
%p A014076 if irem(factorial(2*n), 2*n+1) = 0 then print(2*n+1) end if;
%p A014076 end do: # _Peter Bala_, Jan 24 2017
%t A014076 Select[Range@210, !PrimeQ@ # && OddQ@ # &] (* _Robert G. Wilson v_, Sep 22 2008 *)
%t A014076 Select[Range[1, 199, 2], PrimeOmega[#] != 1 &] (* _Alonso del Arte_, Nov 19 2012 *)
%o A014076 (Haskell)
%o A014076 a014076 n = a014076_list !! (n-1)
%o A014076 a014076_list = filter ((== 0) . a010051) a005408_list
%o A014076 -- _Reinhard Zumkeller_, Sep 30 2011
%o A014076 (PARI) is(n)=n%2 && !isprime(n) \\ _Charles R Greathouse IV_, Nov 24 2012
%o A014076 (Python)
%o A014076 from sympy import primepi
%o A014076 def A014076(n):
%o A014076     if n == 1: return 1
%o A014076     m, k = n-1, primepi(n) + n - 1 + (n>>1)
%o A014076     while m != k:
%o A014076         m, k = k, primepi(k) + n - 1 + (k>>1)
%o A014076     return m # _Chai Wah Wu_, Jul 31 2024
%Y A014076 Cf. A002808, A005408; first differences: A067970, A196274; A047846.
%Y A014076 Cf. A056653.
%K A014076 nonn,easy
%O A014076 1,2
%A A014076 _Warut Roonguthai_