a(n) = customparts(unitary_divisors(n), n)

\\ this one comes from Antti Karttunen. 
unitary_divisors(n) = select(d -> (1==gcd(d, n/d)), divisors(n));

customparts(vparts, n) = {
	maxSum = n; 
	count = 1; 
	
	vpartsCopy = vparts;
	vInv = Map();
	for(i = 1, #vparts, 
		mapput(vInv, vparts[i], i); 
	);
	for(i = 0, n \ vparts[1] - 1,
		custompartsIterate(i)
	);
	
	count
}

custompartsIterate(n) = {
	my(v = vector(n, i, 1), s = sum(i = 1, n, vpartsCopy[v[i]]));
	while(v != 0, 
		j = 0; 
		s = sum(i=1,#v,vpartsCopy[v[i]]);
		if(s < maxSum, 
			mapisdefined(vInv, maxSum - s, &j); 
			if(j >= v[#v],
				count++
			);
			v = nxtVl(v, #vpartsCopy); 	
		,
			v = nxtVs(v)
		)
	)
}

nxtVl(v, {u = oo}) = {
if(v[#v] + 1 <= u, 
v[#v]++; v
,
nxtVs(v)
)
}

nxtVs(v) = {
forstep(j = #v - 1, 1, -1,
 if(v[j] + 1 < v[#v],
 v[j]++;
 for(k = j + 1, #v,
  v[k] = v[j] 
 );
 return(v)
)
) ; return([0])
}