A047845 a(n) = (m-1)/2, where m is the n-th odd nonprime (A014076(n)).

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

Offset: 1

Author

Enoch Haga

Keywords

• easy
• nonn

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 two-dimensional rectangular lattice of unit squares. See A090767 for the three-dimensional 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

Values of k for which A073610(2k+3)=0; values of k for which A061358(2k+3)=0. - Graeme McRae, Jul 18 2006

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

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Slate, Article about USA flag patterns -- this is where Skip Garibaldi gave definitions.

Crossrefs

Complement of A005097.

Programs

Haskell

a047845 = (`div` 2) . a014076  -- Reinhard Zumkeller, Jan 02 2013

Magma

[(n-1)/2 : n in [1..350] | (n mod 2) eq 1 and not IsPrime(n)]; // G. C. Greubel, Oct 16 2023

Maple

for n from 0 to 120 do
    if irem(factorial(2*n), 2*n+1) = 0 then print(n); end if;
end do:
# Peter Bala, Jan 24 2017

Mathematica

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

PARI

print1(0,", ");
forcomposite(n=1,250,if(1==n%2,print1((n-1)/2,", "))); \\ Joerg Arndt, Oct 16 2023

Python

from sympy import primepi
def A047845(n):
    if n == 1: return 0
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m-1>>1 # Chai Wah Wu, Jul 31 2024

SageMath

[(n-1)/2 for n in (1..350) if n%2==1 and not is_prime(n)] # G. C. Greubel, Oct 16 2023

Formula

A193773(a(n)) > 1 for n > 1. - Reinhard Zumkeller, Jan 02 2013

Extensions

Name edited by Jon E. Schoenfield, Oct 16 2023