A339185 a(n) is the least prime p such that the sum of n consecutive primes starting with p has exactly n prime factors, counted with multiplicity, or 0 if no such p exists.

2, 0, 137, 5, 41, 109, 4253, 569, 23057, 821, 405863, 9013, 1049173, 73009, 9742969, 188017, 382355863, 236527, 3198295691, 1843111, 21640201361, 7600499, 376724314301, 33177461, 1974496270177, 305216017, 85571500507397, 148597987, 145412255489161, 951267841, 2609815945304401, 1140850357, 24575914221842531

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 26 2020

Keywords

• nonn

Comments

Conjecture: Such p exists for every n > 2.

Examples

			a(3)=137 because the sum of 3 consecutive primes starting with 137 is 137+139+149=425=5^2*7 is the product of 3 primes counting multiplicity, and 137 is the least prime with this property.
		

Crossrefs

Cf. A001222, A143121, A339269.

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 v[1] 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

A339269(n) = A143121(a(n)+n, a(n)).