New Sum of Powers Log Function

[Namir]

Does any of your previous analysis and approximations work on the following variants of the SopLog summations?

Code:

def sdSum(n,s,x,scale):
  sumx = 1
  factor = 1
  for i in range(n,2,-1):
    sumx += i ** (x * factor)
    factor *= scale
  return sumx - s
  
  
def siSum(n,s,x,scale):
  sumx = 1
  factor = 1
  for i in range(2,n+1):
    sumx += i ** (x * factor)
    factor *= scale
  return sumx - s
  
def siSum2(n,s,x,incr):
  sumx = 1
  factor = 1
  j = 2
  for i in range(2,n):
    sumx += j ** (x * factor)
    j += incr
  return sumx - s
  
  
def siSum3(n,s,x,incr,scale):
  sumx = 1
  factor = 1
  j = 2
  for i in range(2,n+1):
    sumx += j ** (x * factor)
    j += incr
    factor *= scale
  return sumx - s

The "scale" can be above or below 1. The "incr" is an integer >= 1.