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Numbers n such that sum of first n primorials (A143293) is divisible by n.
2

%I #18 Feb 23 2022 11:36:14

%S 1,3,17,51,967,2901,16439,49317,147951,1331559

%N Numbers n such that sum of first n primorials (A143293) is divisible by n.

%C a(5) = 967 is a prime,

%C a(6) = a(5) * 3,

%C a(7) = a(5) * 17,

%C a(8) = a(5) * 51,

%C a(9) = a(5) * 51 * 3,

%C a(10) = a(5) * 51 * 27.

%C The next term, if it exists, is greater than 15600000. - _Alex Ratushnyak_, Jun 16 2013

%e Sum of first 3 primorials is 1+2+6=9, because 9 is divisible by 3, the latter is in the sequence.

%e Sum of first 17 primorials is A143293(17) = 1955977793053588026279. Because A143293(17) 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(5,10000000,6):

%o addPrime(n)

%o addPrime(n+2)

%o sum = 0

%o primorial = n = 1

%o for p in primes:

%o sum += primorial

%o primorial *= p

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

%o n += 1

%o (Python)

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

%o from operator import mul

%o from sympy import prime

%o def A225727_gen(): return (i+1 for i, m in enumerate(accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))) if m % (i+1) == 0)

%o A225727_list = list(islice(A225727_gen(),6)) # _Chai Wah Wu_, Feb 23 2022

%Y Cf. A143293, A002110, A057245, A128981.

%K nonn,hard,more

%O 1,2

%A _Alex Ratushnyak_, May 13 2013