A225728 Primes p such that sum of primorials (A143293) not including p as a factor is divisible by p.

3, 17, 967

Offset: 1

Author

Alex Ratushnyak, May 14 2013

Keywords

• nonn
• bref
• hard
• more

Comments

As in A002110, primorial(0)=1, and primorial(n) = primorial(n-1)*prime(n).

The next term, if it exists, is bigger than 10^8.

Examples

			Sum of primorials not including 3 as a factor is 1 + 2 = 3. Because it's divisible by 3, the latter is in the sequence.
Sum of primorials not including 17 as a factor is 1 + 2 + 6 + 6*5 + 30*7 + 210*11 + 2310*13 = 32589. Because 32589 is divisible by 17, the latter is in the sequence.
		

Crossrefs

Cf. A002110, A143293, A225727.

Programs

PARI

s=P=1;forprime(p=2,1e6,s+=P*=p;if(s%p==0,print1(p", "))) \\ Charles R Greathouse IV, Mar 19 2014

PARI

is(p)=if(!isprime(p),return(0)); my(s=Mod(1,p),P=s); forprime(q=2,p-1,s+=P*=q); s==0 \\ Charles R Greathouse IV, Mar 19 2014

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 = 1
for p in primes:
  sum_ += primorial
  primorial *= p
  if sum_ % p == 0:  print(p, end=', ')

Python

from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A225728_gen(): return (prime(i+1) for i, m in enumerate(accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))) if m % prime(i+1) == 0)
A225728_list = list(islice(A225728_gen(), 3)) # Chai Wah Wu, Feb 23 2022