A288814alt(n) = {
	my(p); 
	if(n < 7, 
		[2 * n - 4, 2 + (n % 2)]
	, 
		p = precprime(n - 2); 
		if(n - p <= 4,
			[(n - p) * p, p]
		,
			my(found = p * A056240alt(n - p)[1], pfound = p, vA = found); 
			while(found / p > n - p && p > 2,
				p = precprime(p - 1); \\print(p" "found / p" "n - p); 
				vA = A056240alt(n - p)[1] * p;
				if(vA < found, 
					found = vA;
					pfound = p; 
				)
				\\ found = min(found, A056240(n - p)[1] * p);
				
			); [found, pfound]
			\\[p * A056240(n - p)[1], p]
			
		)
	)
}

A056240alt(n) = {
	
	if(n <= 5, return([n, n - 2 * (n == 4)])); 
	my(p = precprime(n), found = p * (n - p), vA, pfound);
	if(p == n, 
		return([p, p])
	,
		p = precprime(n - 2); 
		found = p * A056240alt(n - p)[1]; pfound = p; 
		while(found > (n - p) * p && p > 2, 
			p = precprime(p - 1); 
			vA = A056240alt(n - p)[1] * p;
			if(vA < found, 	 
				found = vA;\\min(found, A056240alt(n - p)[1] * p);
				pfound = p; 
			)
		); 
		[found, pfound]  
	) 
}

a(n) = {my(sp = prime(n + 1), t = 0);
sp = nextprime(t * 10^6);  
forprime(p = sp, , 
		if(p > 10^6 * t, print(10^6 * t); t++); 
	 	
		x = A288814alt(p); 
		fmax = x[2];
		
		if (dist(fmax, p, n) == n && x[1] % fmax == 0, 
			y = fmax * A056240alt(p - fmax)[1]; 
			if (y == x[1], 
				return (p); 
			) 
		) 
	 
) 
}

first(n) = {my(sp = prime(n + 1), res = vector(n), todo = n);
forprime(p = 4, , 
		x = A288814alt(p); 
		fmax = x[2];
		d = dist(fmax, p, n);
		if(x[1] % fmax == 0,
			if(d <= n, 
		    		y = fmax * A056240alt(p - fmax)[1];
				if(y == x[1] && res[d] == 0,
					res[d] = p; todo--; 
					print(sig(d, p));
					
					if(todo == 0, return(res)); 
					
				)
			,
				res = concat(res, vector(d - #res)); 
				y = fmax * A056240alt(p - fmax)[1];
				if(y == x[1] && res[d] == 0,
					res[d] = p;
					print(sig(d, p));
					
				)		
			)
		)
) 
}


sig(n, an) = {my(res = "a(", p, can); 
res = concat(res, n);
res = concat(res, ") = ");
res = concat(res, an); 
res = concat(res, " ~ "); 
res = concat(res, n); 
res = concat(res, "(");
can = precprime(an - 1);
res = concat(res, an - can); 
an = can;
for(i = 2, n, 
	can = precprime(an - 1);
	res = concat(res, ", ");
	res = concat(res, an - can); 
	an = can; 
); res = concat(res, "),"); res

}

\\return distance between primes p1 and p2 with indices i1 and i2; the difference between indices of prime; primepi(p2) - primepi(p1). if p1 > p2 then it's 0. 
\\ search for a maximum distance of n + 1; if i2 - i1 > n we say it's n + 1. 
dist(p1, p2, n) = {my(res = 0);
	forprime(p = p1 + 1, p2, 
		res++; if(res > n, return(n + 1)); 
	); res
}