OFFSET
1,1
COMMENTS
Numbers of the form F(K, L) = KL+(K-1)(L-1), K, L > 1, i.e. 2KL - (K+L) + 1, sorted and duplicates removed.
If K=1, L=1 were allowed, this would contain all positive integers.
Positive numbers > 1 but not of the form (odd primes plus one)/2. - Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003
In other words, numbers n such that 2n-1, or equally, A064216(n) is a composite number. - Antti Karttunen, Apr 17 2015
Note: the following comment was originally applied in error to the numerically similar A246371. - Allan C. Wechsler, Aug 01 2022
From Matthijs Coster, Dec 22 2014: (Start)
Also area of (over 45 degree) rotated rectangles with sides > 1. The area of such rectangles is 2ab - a - b + 1 = 1/2((2a-1)(2b-1)+1).
Example: Here a = 3 and b = 5. The area = 23.
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(End)
The smallest integer > k/2 and coprime to k, where k is the n-th odd composite number. - Mike Jones, Jul 22 2024
Numbers k such that A193773(k-1) > 1. - Allan C. Wechsler, Oct 22 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A008508(n) + n + 1.
From Antti Karttunen, Apr 17 2015: (Start)
a(n) = n + A000720(A071904(n)). [The above formula reduces to this. A000720(k) gives number of primes <= k, and A071904 gives the n-th odd composite number.]
a(n) = A104275(n+1). (End)
a(n) = A047845(n+1)+1. - Amiram Eldar, Jul 30 2024
PROG
(Scheme) ;; with Antti Karttunen's IntSeq-library.
(define A053726 (MATCHING-POS 1 1 (lambda (n) (and (> n 1) (not (prime? (+ n n -1)))))))
;; Antti Karttunen, Apr 17 2015
(Scheme) ;; with Antti Karttunen's IntSeq-library.
;; Antti Karttunen, Apr 17 2015
(Python)
from sympy import isprime
def ok(n): return n > 1 and not isprime(2*n-1)
print(list(filter(ok, range(1, 117)))) # Michael S. Branicky, May 08 2021
(Python)
from sympy import primepi
def A053726(n):
if n == 1: return 5
m, k = n, (r:=primepi(n)) + n + (n>>1)
while m != k:
m, k = k, (r:=primepi(k)) + n + (k>>1)
return r+n # Chai Wah Wu, Aug 02 2024
(PARI) select( {is_A053726(n)=n>4 && !isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dan Asimov, asimovd(AT)aol.com, Apr 09 2003
EXTENSIONS
More terms from Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003
STATUS
approved