A225728 - OEIS

%I #20 Feb 24 2022 09:02:08

%S 3,17,967

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

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

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

%e 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.

%e 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.

%o (Python)

%o primes = [2]*2

%o primes[1] = 3

%o def addPrime(k):

%o   for p in primes:

%o     if k%p==0:  return

%o     if p*p > k:  break

%o   primes.append(k)

%o for n in range(5,100000000,6):

%o   addPrime(n)

%o   addPrime(n+2)

%o sum = 0

%o primorial = 1

%o for p in primes:

%o   sum += primorial

%o   primorial *= p

%o   if sum % p == 0:  print p,

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

%o (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

%o (Python)

%o from itertools import chain, accumulate, count, islice

%o from operator import mul

%o from sympy import prime

%o 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)

%o A225728_list = list(islice(A225728_gen(), 3)) # _Chai Wah Wu_, Feb 23 2022

%Y Cf. A002110, A143293, A225727.

%K nonn,bref,hard,more

%O 1,1

%A _Alex Ratushnyak_, May 14 2013