A253975 -id:A253975 - cp's OEIS frontend

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

Alex Ratushnyak, Feb 03 2015

nonn

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

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

Cf. A253975.

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

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 *)

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

A255361 Number of ways n can be represented as x*y+x+y where x>=y>1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 3, 0, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 0, 0, 4, 0, 0, 2, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 0, 2, 1, 1, 2, 0, 3, 2, 0, 0, 4, 1, 0, 1, 2, 0, 4, 1, 1, 1, 0, 1, 4, 0, 1, 2, 3, 0, 2, 0, 2, 3, 0

Offset: 0

Alex Ratushnyak, Feb 21 2015

nonn

			8 = 2*2 + 2 + 2, this is the only representation, so a(8)=1.
23 = 2*7 + 2 + 7 = 3*5 + 3 + 5, two representations, so a(23)=2.

Antti Karttunen, Table of n, a(n) for n = 0..3300

Cf. A253975, A254636.

a[n_] := (r = Reduce[x >= y > 1 && n == x*y + x + y, {x, y}, Integers]; Which[r[[0]] === And, 1, r[[0]] === Or, Length[r], True, 0]);
Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Jan 23 2018 *)

a(n) = {nb = 0; for (y=2, n\2, for (x=y, n\2, nb += ((x*y+x+y) == n););); nb;} \\ Michel Marcus, Feb 22 2015

TOP = 1000
a = [0]*TOP
for y in range(2, TOP//2):
    for x in range(y, TOP//2):
        k = x*y + x + y
        if k>=TOP: break
        a[k]+=1
print(a)

from sympy import divisor_count
def A255361(n): return int((divisor_count(n+1)-1>>1)-(n&1)) if n!=1 else 0 # Chai Wah Wu, Oct 15 2024

Let d = A000005; then a(n) = floor((d(n+1) - 1)/2) for even n and a(n) = floor((d(n+1) - 3) / 2) for odd n>1. - Ivan Neretin, Sep 07 2015

More terms from Antti Karttunen, Sep 22 2017

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

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A255361 Number of ways n can be represented as x*y+x+y where x>=y>1.

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