efficient generation of restricted divisors

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This question is in the same venue as my previous one on generation of unitary divisors.

This time I'm interested in generating all divisors of a given factored integer that are below a given bound B. For simplicity I will focus on just counting such divisors.

A straightforward way is to use the divisors() function:

import itertools
def divbelow_1(n,B):
    return sum(1 for _ in itertools.takewhile(lambda d: d<=B, divisors(n)))

The downside is that this function constructs a list of all divisors of $n$ (there are $\tau(n)$ of them), which limits its applicability. Still, if this list fits into the memory, divbelow_1() works quite fast. I'm interested in the case when $\tau(n)$ is large, but the number of divisors below B is moderate. Ideally, I'd like to have a function that have comparable performance to divbelow_1(n,B) when B==n, and that at the same time avoids generation of divisors above B.

My attempt was:

# N = list(factor(n)), a list of pairs (prime,exponent) forming the factorization of n 
def divbelow_2(N,B,i=0):
    if B <= 1:
        return B
    if i>=len(N):
        return 1

    p,e = N[i]
    result = 0
    for k in (0..e):
        result += divbelow_2(N,B,i+1)
        B //= p
        if B==0:
            break

    return result

White it looks quite efficient for me from theoretical perspective, it loses badly to divbelow_1() in practice. Here is a particular example:

sage: %time divbelow_1(47987366728745697656468743117440000, 61448226992991993)
CPU times: user 4.73 s, sys: 368 ms, total: 5.1 s
Wall time: 5.1 s
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sage: %time divbelow_2(list(factor(47987366728745697656468743117440000)), 61448226992991993)
CPU times: user 1min 31s, sys: 7.89 ms, total: 1min 31s
Wall time: 1min 31s
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I can cope with 2-3 fold slowdown, but not with 18-fold one. So, my question is:

Q: Why divbelow_2() is so slow, and if there is there a room for improvement? (without using divisors())

ADDED. I suspect the drastic slowdown is caused by the recursion overhead. Here is a non-recursive version that generates the list of all divisors below B:

def divbelow_7(n,B):
    div = [1]
    for p, e in factor(n):
        ext = div
        for i in range(e):
            ext = [r for d in ext if (r := d * p) <= B]
            div.extend(ext)
    return len(div)

On the above example it even beats divbelow_1():

sage: %time divbelow_7(47987366728745697656468743117440000, 61448226992991993)
CPU times: user 1.97 s, sys: 176 ms, total: 2.15 s
Wall time: 2.15 s
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