A146208 a(n) is the number of arithmetic progressions of 2 or more integers with product = n.

4, 6, 6, 4, 10, 4, 11, 8, 10, 4, 12, 4, 8, 12, 12, 4, 12, 4, 12, 10, 8, 4, 26, 6, 8, 9, 14, 4, 16, 4, 13, 8, 8, 10, 20, 4, 8, 8, 20, 4, 18, 4, 12, 16, 8, 4, 26, 6, 12, 8, 12, 4, 16, 10, 16, 8, 8, 4, 26, 4, 8, 14, 19, 8, 18, 4, 12, 8, 16, 4, 24, 4, 8, 12

Offset: 2

Author

Naohiro Nomoto, Oct 28 2008

Keywords

• easy
• nonn

Comments

a(n)=number of all integer triples (x,y,z) such that Product_{k=0..z} (x + (y*k)) = n, where n>1, z>0.

Examples

			a(8) = 11 as we can have
 (x=-8,y=7,z=1; -8 * -1),
 (x=-4,y=2,z=1; -4 * -2),
 (x=-4,y=3,z=2; -4 * -1 * 2),
 (x=-2,y=-2,z=1; -2 * -4),
 (x=-1,y=-7,z=1; -1 * -8),
 (x=1,y=7,z=1; 1 * 8),
 (x=2,y=-3,z=2; 2 * -1 * -4),
 (x=2,y=0,z=2; 2 * 2 * 2),
 (x=2,y=2,z=1; 2 * 4),
 (x=4,y=-2,z=1; 4 * 2),
 (x=8,y=-7,z=1; 8 * 1). - Example added by _Antti Karttunen_, Feb 28 2023
a(9) = 8 as we can have
 (x=-3,y=0,z=1; -3 * -3),
 (x=3,y=0,z=1; 3 * 3),
 (x=-9,y=8,z=1; -9 * -1),
 (x=1,y=8,z=1; 1 * 9),
 (x=-1,y=-8,z=1; -1 * -9),
 (x=9,y=-8,z=1; 9 * 1),
 (x=3,y=-2,z=3; 3 * 1 * -1 * -3),
 (x=-3,y=2,z=3; -3 * -1 * 1 * 3).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A062011, A361015.

Programs

PARI

A146208(n) = sum(x=-n,n,sum(y=-n,n,sum(z=1,n,n==prod(k=0,z,x+(y*k))))); \\ (Slow!) - Antti Karttunen, Feb 28 2023

Python

from sympy import divisors
def A146208(n):
    ds = divisors(n)
    c, s = 0, [-d for d in ds[::-1]]+ds
    for x in s:
        d2 = [d//x for d in ds if d%x==0]
        for y in (f-x for f in [-d for d in d2[::-1]]+d2):
            m, k = x*(z:=x+y), 1
            while n >= abs(m) and k<=n:
                if n == m:
                    c += 1
                z += y
                m *= z
                k += 1
    return c # Chai Wah Wu, May 11 2023

Formula

a(n) = A062011(n) + A361015(n). - Antti Karttunen, Feb 28 2023