A339269 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.

2, 0, 425, 36, 243, 756, 29889, 4704, 207765, 8448, 4465125, 108864, 13640319, 1022976, 146146275, 3010560, 6500054871, 4259840, 60767621145, 36864000, 454444233597, 167215104, 8664659236485, 796262400, 49362406764957, 7935623168, 2310430513712625, 4160749568, 4216955409197811, 28538044416

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 29 2020

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for every n > 2.

If n > 2 is odd, the sum of n consecutive odd primes is odd, so (if nonzero) a(n) >= 3^n.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A001222, A143121, A339185.

Programs

Maple

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]);

Formula

a(n) = A143121(A339185(n)+n, A339185(n)).