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Robert Israel, <a href="/A339269/b339269.txt">Table of n, a(n) for n = 1..50</a>
If n > 2 is odd, the sum of n consecutive odd primes is odd, so (if nonzero) a(n) >= 3^n.
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a(n) is the least number that is the product of n primes (not necessarily distinct) and is the sum of n consecutive primes, or 0 if there are none.
sumofconsecprimes:= proc(x, n)
local P, k, p, q, t;
P:= nextprime(floor(x/n));
p:= P; q:= P;
for k from 1 to n-1 do
if k::even or q = 2 then p:= nextprime(p); P:= P, p;
else q:= prevprime(q); P:= q, P;
fi
od;
P:= [P];
t:= convert(P, `+`);
if t = x then return P fi;
if t > x then
while t > x do
if q = 2 then return false fi;
q:= prevprime(q);
t:= t + q - p;
P:= [q, op(P[1..-2])];
p:= P[-1];
if t = x then return P fi;
od
else
while t < x do
p:= nextprime(p);
t:= t + p - q;
P:= [op(P[2..-1]), p];
q:= P[1];
if t = x then return P fi;
od
fi;
false
end proc:
children:= proc(r) local L, x, p, q, t, R;
x:= r[1];
L:= r[2];
t:= L[-1];
p:= t[1]; q:= nextprime(p);
if t[2]=1 then t:= [q, 1];
else t:= [p, t[2]-1], [q, 1]
fi;
R:= [x*q/p, [op(L[1..-2]), t]];
if nops(L) >= 2 then
p:= L[-2][1];
q:= L[-1][1];
if L[-2][2]=1 then t:= [q, L[-1][2]+1]
else t:= [p, L[-2][2]-1], [q, L[-1][2]+1]
fi;
R:= R, [x*q/p, [op(L[1..-3]), t]]
fi;
[R]
end proc:
f:= proc(n) local Q, t, x, v;
uses priqueue;
initialize(Q);
if n::even then insert([-2^n, [[2, n]]], Q)
else insert([-3^n, [[3, n]]], Q)
fi;
do
t:= extract(Q);
x:= -t[1];
v:= sumofconsecprimes(x, n);
if v <> false then return x fi;
for t in children(t) do insert(t, Q) od;
od
end proc:
f(1):= 2:
f(2):= 0:
map(f, [$1..34]);
0,1,1
Conjecture: a(n) > 0 for every n > 2.
a(3)=425 because 425 = 5^2*7 is the product of three primes and 425 = 137+139+149 is the sum of three consecutive primes, and no smaller number has this property.