A345976 a(n) is the least sum of n consecutive primes that has exactly n distinct prime divisors.

2, 12, 1015, 390, 26565, 66990, 20722065, 49519470, 14809429635, 14908423530, 8397151047285, 20308367048970, 18572001666283065, 58399604637033660, 40356959620833100245, 80449728529316576430, 159773203138878243869955, 270976605209664381237210, 813028675159280138982227805

Offset

1,1

Links

Table of n, a(n) for n=1..19.

Example

a(2) = 12 because 5 (= A345977(2)) + 7 = 12 = 2^2*3 is the first sum S of 2 consecutive primes for which omega(S) = 2. 2 + 3 = 5, and 3 + 5 = 8 = 2^3 both have only one prime dividing S.
a(3) = 1015 = 331 + 337 + 347 = 5*7*29;
a(4) = 390 = 89 + 97 + 101 + 103 = 2*3*5*13;
a(5) = 26565 = 5297 + 5303 + 5309 + 5323 + 5333 = 3*5*7*11*23.
From Jon E. Schoenfield, Sep 23 2023: (Start)
Since a(n) is the sum of n consecutive primes, a(n) is even iff n is even (provided that 2 is not among the consecutive primes, which happens only at n=1).
It seems that a(n) is usually squarefree, but a(2) and a(14) are multiples of 4.
The prime factorizations of the first 19 terms are shown in the table below. (To highlight the tendency of the terms to include the smallest odd primes among their divisors, each prime < 80 has its own column.)
.
   n  prime factorization of a(n)                        primes < 80 | > 80
  --  ---------------------------------------------------------------+-----
   1    2                                                            |
   2  2^2*3                                                          |
   3        5*7               *29                                    |
   4    2*3*5     *13                                                |
   5      3*5*7*11         *23                                       |
   6    2*3*5*7*11            *29                                    |
   7      3*5*7   *13*17*19                  *47                     |
   8    2*3*5*7*11*13*17                                             |*97
   9      3*5*7*11*13   *19*23      *37               *61            |
  10    2*3*5*7*11*13*17*19   *29               *53                  |
  11      3*5*7*11*13*17   *23*29*31*37   *43                        |
  12    2*3*5*7*11*13*17*19*23*29         *43                  *73   |
  13      3*5*7*11*13*17*19*23*29*31      *43      *59         *73   |
  14  2^2*3*5*7*11*13*17*19*23*29*31   *41      *53      *67         |
  15      3*5*7*11*13*17*19*23*29*31   *41*43   *53*59         *73   |
  16    2*3*5*7*11*13*17*19*23*29*31*37   *43   *53      *67*71      |
  17      3*5*7*11*13*17*19*23*29*31*37*41*43   *53*59         *73   |*107
  18    2*3*5*7*11*13*17*19*23   *31*37*41*43*47*53*59*61*67         |
  19      3*5*7*11*13*17*19*23*29*31*37*41*43*47   *59*61   *71   *79|*131
(End)

Prog

(Python)
from sympy import nextprime, prime, primefactors, primerange
def a(n):
    plst = [p for p in primerange(1, prime(n)+1)]
    while len(primefactors(sum(plst))) != n:
        plst = plst[1:] + [nextprime(plst[-1])]
    return sum(plst)
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Jul 16 2021

Crossrefs

A345977 provides the first prime in the sum.

Cf. A001221, A346382.

Sequence in context: A112373 A058975 A057120 * A112512 A006751 A023989

Adjacent sequences: A345973 A345974 A345975 * A345977 A345978 A345979

Keyword

nonn

Author

Hugo Pfoertner, Jul 16 2021

Extensions

a(8)-a(10) from Michael S. Branicky, Jul 16 2021

a(11)-a(12) from Martin Ehrenstein, Jul 17 2021

a(13)-a(19) from Jon E. Schoenfield, Sep 23 2023

Status

approved