A225727 Numbers k such that sum of first k primorials (A143293) is divisible by k.

1, 3, 17, 51, 967, 2901, 16439, 49317, 147951, 1331559

Offset: 1

Author

Alex Ratushnyak, May 13 2013

Keywords

• nonn
• hard
• more

Comments

a(5) = 967 is a prime,

a(6) = a(5) * 3,

a(7) = a(5) * 17,

a(8) = a(5) * 51,

a(9) = a(5) * 51 * 3,

a(10) = a(5) * 51 * 27.

The next term, if it exists, is greater than 15600000. - Alex Ratushnyak, Jun 16 2013

Examples

			Sum of first 3 primorials is 1+2+6=9, because 9 is divisible by 3, the latter is in the sequence.
Sum of first 17 primorials is A143293(17) = 1955977793053588026279. Because A143293(17) is divisible by 17, the latter is in the sequence.
		

Crossrefs

Cf. A143293, A002110, A057245, A128981.

Programs

Python

primes = [2,3]
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5,1000000,6):
  addPrime(n)
  addPrime(n+2)
sum_ = 0
primorial = n = 1
for p in primes:
  sum_ += primorial
  primorial *= p
  if sum_ % n == 0:  print(n, end=', ')
  n += 1

Python

from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A225727_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))) if m % (i+1) == 0)
A225727_list = list(islice(A225727_gen(),6)) # Chai Wah Wu, Feb 23 2022