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%I #74 Oct 20 2023 06:47:54
%S 0,4,7,10,12,13,16,17,19,22,24,25,27,28,31,32,34,37,38,40,42,43,45,46,
%T 47,49,52,55,57,58,59,60,61,62,64,66,67,70,71,72,73,76,77,79,80,82,84,
%U 85,87,88,91,92,93,94,97,100,101,102,103,104,106,107,108,109,110,112,115
%N a(n) = (m-1)/2, where m is the n-th odd nonprime (A014076(n)).
%C Also (starting with 2nd term) numbers of the form 2xy+x+y for x and y positive integers. This is also the numbers of sticks needed to construct a two-dimensional rectangular lattice of unit squares. See A090767 for the three-dimensional generalization. - _John H. Mason_, Feb 02 2004
%C Note that if k is not in this sequence, then 2*k+1 is prime. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
%C Values of k for which A073610(2k+3)=0; values of k for which A061358(2k+3)=0. - _Graeme McRae_, Jul 18 2006
%C This sequence also arises in the following way: take the product of initial odd numbers, i.e., the product (2n+1)!/(n!*2^n) and factor it into prime numbers. The result will be of the form 3^f(3)*5^f(5)*7^f(7)*11^f(11)... . Then f(3)/f(5) = 2, f(3)/f(7) = 3, f(3)/f(11) = 5, ... and this sequence forms (for sufficiently large n, of course) the sequence of natural numbers without 4,7,10,12,..., i.e., these numbers are what is lacking in the present sequence. - Andrzej Staruszkiewicz (uszkiewicz(AT)poczta.onet.pl), Nov 10 2007
%C Also "flag short numbers", i.e., number of dots that can be arranged in successive rows of K, K+1, K, K+1, K, ..., K+1, K (assuming there is a total of L > 1 rows of size K > 0). Adapting Skip Garibaldi's terms, sequence A053726 would be "flag long numbers" because those patterns begin and end with the long lines. If you convert dots to sticks, you get the lattice that John H. Mason mentioned. - _Juhani Heino_, Oct 11 2014
%C Numbers k such that (2*k)!/(2*k + 1) is an integer. - _Peter Bala_, Jan 24 2017
%C Except for a(1)=0: numbers of the form k == j (mod 2j+1), j >= 1, k > 2j+1. - _Bob Selcoe_, Nov 07 2017
%H Reinhard Zumkeller, <a href="/A047845/b047845.txt">Table of n, a(n) for n = 1..10000</a>
%H Slate, <a href="http://www.slate.com/articles/life/do_the_math/2010/06/13_stripes_and_51_stars.html">Article about USA flag patterns</a> -- this is where Skip Garibaldi gave definitions.
%F A193773(a(n)) > 1 for n > 1. - _Reinhard Zumkeller_, Jan 02 2013
%p for n from 0 to 120 do
%p if irem(factorial(2*n), 2*n+1) = 0 then print(n); end if;
%p end do:
%p # _Peter Bala_, Jan 24 2017
%t (Select[Range[1, 231, 2], PrimeOmega[#] != 1 &] - 1)/2 (* _Jayanta Basu_, Aug 11 2013 *)
%o (Haskell)
%o a047845 = (`div` 2) . a014076 -- _Reinhard Zumkeller_, Jan 02 2013
%o (Magma) [(n-1)/2 : n in [1..350] | (n mod 2) eq 1 and not IsPrime(n)]; // _G. C. Greubel_, Oct 16 2023
%o (SageMath) [(n-1)/2 for n in (1..350) if n%2==1 and not is_prime(n)] # _G. C. Greubel_, Oct 16 2023
%o (PARI) print1(0,", ");
%o forcomposite(n=1,250,if(1==n%2,print1((n-1)/2,", "))); \\ _Joerg Arndt_, Oct 16 2023
%Y Complement of A005097.
%K easy,nonn
%O 1,2
%A _Enoch Haga_
%E Name edited by _Jon E. Schoenfield_, Oct 16 2023
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