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%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
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