A254636 Numbers that cannot be represented as x*y + x + y, where x>=y>1.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16, 18, 21, 22, 25, 28, 30, 33, 36, 37, 40, 42, 45, 46, 52, 57, 58, 60, 61, 66, 70, 72, 73, 78, 81, 82, 85, 88, 93, 96, 100, 102, 105, 106, 108, 112, 117, 121, 126, 130, 133, 136, 138, 141, 145, 148, 150, 156, 157, 162, 165, 166, 172

Offset: 1

Author

Alex Ratushnyak, Feb 03 2015

Keywords

• nonn

Comments

0, 7 and numbers n such that n+1 is either prime or twice a prime. - Robert Israel, Aug 05 2015

Crossrefs

Cf. A091529 (appears to be essentially the same, except first few terms).

Cf. A253975.

Programs

Maple

sort([0,7, op(select(t -> isprime(t+1), [$1..10^4])), op(select(t -> isprime((t+1)/2),[2*i+1$i=1..5*10^3]))]); # Robert Israel, Aug 05 2015

Mathematica

r[n_] := Reduce[x >= y > 1 && n == x y + x + y, {x, y}, Integers];
Reap[For[n = 0, n <= 200, n++, If[r[n] === False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 28 2019 *)

Python

from sympy import primepi
def A254636(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x-(x>=7)-primepi(x+1)-primepi(x+1>>1))
    return bisection(f,n-1,n-1) # Chai Wah Wu, Oct 14 2024