A053726 "Flag numbers": 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 > 1).

5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 71, 72, 73, 74, 77, 78, 80, 81, 83, 85, 86, 88, 89, 92, 93, 94, 95, 98, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 116

Offset: 1

Author

Dan Asimov, asimovd(AT)aol.com, Apr 09 2003

Keywords

• nonn
• easy

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

Crossrefs

Essentially same as A104275, but without the initial one.

A144650 sorted into ascending order, with duplicates removes.

Cf. A006254 (complement, apart from 1, which is in neither sequence).

Cf. also A000720, A008508, A064216, A071904, A199593, A250474.

Differs from its subsequence A246371 for the first time at a(8) = 20, which is missing from A246371.

Programs

PARI

select( {is_A053726(n)=n>4 && !isprime(n*2-1)}, [1..115]) \\ M. F. Hasler, Aug 02 2022

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

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.
(define (A053726 n) (+ n (A000720 (A071904 n))))
;; Antti Karttunen, Apr 17 2015

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) = A116922(A071904(n)). - Mike Jones, Jul 22 2024

a(n) = A047845(n+1)+1. - Amiram Eldar, Jul 30 2024

Extensions

More terms from Douglas Winston (douglas.winston(AT)srupc.com), Sep 11 2003