

A047845


a(n) = (m1)/2, where m is the nth odd nonprime (A014076(n)).


31



0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, 40, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 84, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115
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OFFSET

1,2


COMMENTS

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 twodimensional rectangular lattice of unit squares. See A090767 for the threedimensional generalization.  John H. Mason, Feb 02 2004
Note that if k is not in this sequence, then 2*k+1 is prime.  Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
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
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
Numbers k such that (2*k)!/(2*k + 1) is an integer.  Peter Bala, Jan 24 2017
Except for a(1)=0: numbers of the form k == j (mod 2j+1), j >= 1, k > 2j+1.  Bob Selcoe, Nov 07 2017


LINKS



FORMULA



MAPLE

for n from 0 to 120 do
if irem(factorial(2*n), 2*n+1) = 0 then print(n); end if;
end do:


MATHEMATICA

(Select[Range[1, 231, 2], PrimeOmega[#] != 1 &]  1)/2 (* Jayanta Basu, Aug 11 2013 *)


PROG

(Haskell)
(Magma) [(n1)/2 : n in [1..350]  (n mod 2) eq 1 and not IsPrime(n)]; // G. C. Greubel, Oct 16 2023
(SageMath) [(n1)/2 for n in (1..350) if n%2==1 and not is_prime(n)] # G. C. Greubel, Oct 16 2023
(PARI) print1(0, ", ");
forcomposite(n=1, 250, if(1==n%2, print1((n1)/2, ", "))); \\ Joerg Arndt, Oct 16 2023


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



