A254671 Numbers that can be represented as x*y + x + y, where x >= y > 1.

8, 11, 14, 15, 17, 19, 20, 23, 24, 26, 27, 29, 31, 32, 34, 35, 38, 39, 41, 43, 44, 47, 48, 49, 50, 51, 53, 54, 55, 56, 59, 62, 63, 64, 65, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 107, 109, 110, 111, 113

Offset: 1

Author

Alex Ratushnyak, Feb 04 2015

Keywords

• nonn

Comments

Apparently 8 and the elements of A061743. - R. J. Mathar, Feb 19 2015

This is true. For proof, see link.

As x*y + x + y = (x + 1)*(y + 1) - 1 where x >= y > 1 we have k = a(n) is in the sequence if and only if k + 1 is an odd composite or k + 1 is an even number with more than 4 divisors. - David A. Corneth, Oct 15 2024

Examples

			14 = 2*4 + 2 + 4.
15 = 3*3 + 3 + 3.
There is no way to express 16 in this form, so it is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, A254571 and A061743

Crossrefs

Cf. A071904, A254636 is the complement.

Programs

Maple

filter:= proc(n) local t;
  if n::odd then numtheory:-tau(n+1) > 4 else not isprime(n+1) fi
end proc:
select(filter, [$1..200]); # Robert Israel, Nov 14 2024

Mathematica

sol[t_] := Solve[x >= y > 1 && x y + x + y == t, {x, y}, Integers];
Select[Range[200], sol[#] != {}&] (* Jean-François Alcover, Jul 28 2020 *)

Python

def aupto(limit):
    cands = range(2, limit//3+1)
    nums = [x*y+x+y for i, y in enumerate(cands) for x in cands[i:]]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(113)) # Michael S. Branicky, Aug 11 2021

Python

from sympy import primepi
def A254671(n):
    def f(x): return int(n+(x>=7)+primepi(x+1)+primepi(x+1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 14 2024