cp's OEIS Frontend

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

%I A225728 #24 Jul 04 2025 11:15:05
%S A225728 3,17,967
%N A225728 Primes p such that sum of primorials (A143293) not including p as a factor is divisible by p.
%C A225728 As in A002110, primorial(0)=1, and primorial(n) = primorial(n-1)*prime(n).
%C A225728 The next term, if it exists, is bigger than 10^8.
%e A225728 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 A225728 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 A225728 (Python)
%o A225728 primes = [2,3]
%o A225728 def addPrime(k):
%o A225728   for p in primes:
%o A225728     if k%p==0:  return
%o A225728     if p*p > k:  break
%o A225728   primes.append(k)
%o A225728 for n in range(5,1000000,6):
%o A225728   addPrime(n)
%o A225728   addPrime(n+2)
%o A225728 sum_ = 0
%o A225728 primorial = 1
%o A225728 for p in primes:
%o A225728   sum_ += primorial
%o A225728   primorial *= p
%o A225728   if sum_ % p == 0:  print(p, end=', ')
%o A225728 (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 A225728 (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 A225728 (Python)
%o A225728 from itertools import chain, accumulate, count, islice
%o A225728 from operator import mul
%o A225728 from sympy import prime
%o A225728 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 A225728_list = list(islice(A225728_gen(), 3)) # _Chai Wah Wu_, Feb 23 2022
%Y A225728 Cf. A002110, A143293, A225727.
%K A225728 nonn,bref,hard,more
%O A225728 1,1
%A A225728 _Alex Ratushnyak_, May 14 2013