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

2,2

Comments

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

Links

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

Formula

a(n) = A146208(n) - A062011(n).

Example

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).

Prog

(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

Crossrefs

Cf. A062011, A146208.

Sequence in context: A291900 A263146 A365047 * A028959 A317642 A258762

Adjacent sequences: A361012 A361013 A361014 * A361016 A361017 A361018

Keyword

nonn

Author

Antti Karttunen, Feb 28 2023

Status

approved