A361015 Number of arithmetic progressions of 3 or more integers whose product is equal to n.

0, 2, 0, 0, 2, 0, 3, 2, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 5, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20

Offset: 2

Author

Antti Karttunen, Feb 28 2023

Keywords

• nonn

Comments

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

Examples

			a(3) = 2 as we have solutions (x=-3,y=2,z=2; -3 * -1 * 1) and (x=1,y=-2,z=2; 1 * -1 * -3).
a(8) = 3 as we have solutions  (x=-4,y=3,z=2; -4 * -1 * 2), (x=2,y=-3,z=2; 2 * -1 * -4), and (x=2,y=0,z=2; 2*2*2).
a(27) = 1 as we have a unique solution (x=3,y=0,z=2; 3*3*3).
a(32) = 1 as we have a unique solution (x=2,y=0,z=4; 2*2*2*2*2).
a(64) = 5 as we have solutions (x=-8,y=6,z=2; -8 * -2 * 4), (x=-2,y=0,z=5; (-2)^6), (x=2,y=0,z=5; 2^6), (x=4,y=-6,z=2; 4 * -2 * -8), and (x=4,y=0,z=2; 4*4*4).
a(81) = 4 as we have solutions (x=-9,y=6,z=2; -9 * -3 * 3), (x=-3,y=0,z=3; -3 * -3 * -3 * -3), (x=3,y=-6,z=2; 3 * -3 * -9), and (x=3,y=0,z=3; 3*3*3*3).
a(300) = 2 as we have solutions (x=-25,y=13,z=2; -25 * -12 * 1) and (x=1,y=-13,z=2; 1 * -12 * -25).
		

Links

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

Crossrefs

Cf. A062011, A146208.

Programs

PARI

A361015(n) = sum(x=-n,n,sum(y=-n,n,sum(z=2,n,n==prod(k=0,z,x+(y*k))))); \\ (Slow!)

Python

from sympy import divisors
def A361015(n):
    ds = divisors(n)
    c, s = -len(ds)<<1, [-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) = A146208(n) - A062011(n).