A225841 Numbers n such that the sum of first n primorial numbers is divisible by n.

1, 2, 4, 523, 1046, 2092

Offset: 1

Author

Alex Ratushnyak, May 21 2013

Keywords

• nonn
• hard
• more

Comments

The k-th primorial number is defined as the product of the first k primes.

The next term, if it exists, is greater than 14000000. - Alex Ratushnyak, Jun 13 2013

If a prime p | a(n) for some n, then p = 2, p = 523, or p > 10^8. Any such prime is itself a member of this sequence. From this (and a small amount of additional calculation) it follows that any other terms below 10^10 are of the form 2^k * p for p > 10^8. - Charles R Greathouse IV, Feb 09 2014

Examples

			2 + 2*3 + 2*3*5 + 2*3*5*7 = 2 + 6 + 30 + 210 = 248, because 248 is divisible by 4, the latter is in the sequence.
		

Crossrefs

Cf. A060389, A002110, A045345, A143293, A225727.

Programs

Mathematica

With[{nn=2100},Select[Thread[{Accumulate[FoldList[Times,Prime[ Range[ nn]]]],Range[nn]}],Divisible[ #[[1]],#[[2]]]&]][[All,2]] (* Harvey P. Dale, Jul 29 2021 *)

PARI

list(maxx)={n=prime(1); cnt=1;summ=0;scnt=0;
while(n<=maxx,summ=summ+prodeuler(x=1,prime(cnt),x);
if(summ%cnt==0,scnt++;print(scnt,"  ",cnt) );cnt++;n=nextprime(n+1) ); }
\\note MUST increase precision to 10000+ digits \\Bill McEachen, Feb 04 2014

PARI

P=1;S=n=0;forprime(p=2,1e4,S+=P*=p;if(S%n++==0,print1(n", "))) \\ Charles R Greathouse IV, Feb 05 2014

PARI

is(n)=my(q=prime(n),P=Mod(1,n),S);forprime(p=2,q,S+=P*=p);!S \\ Charles R Greathouse IV, Feb 05 2014

Python

primes = []
n = 1
sum = 2
primorial = 6
def addPrime(k):
  global n, sum, primorial
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  sum += primorial
  primorial *= k
  n += 1
  if sum % n == 0:  print(n, end=',')
print(1, end=',')
for p in range(5, 100000, 6):
  addPrime(p)
  addPrime(p+2)

Python

from itertools import accumulate, count, islice
from operator import mul
from sympy import prime
def A225841_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate((prime(n) for n in count(1)), mul))) if m % (i+1) == 0)
A225841_list = list(islice(A225841_gen(),6)) # Chai Wah Wu, Feb 23 2022