A235431 The smallest positive number that must be added to or subtracted from the sum of the first n primes in order to get a prime.

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 4, 11, 2, 1, 6

Offset: 1

Author

R. J. Cano, Jan 17 2014

Keywords

• nonn
• easy
• hear

Comments

The primes in A013918 would have associated a(n)=0 if not for the qualifier "positive" in the definition.

The sum of the first n primes appears to be close to a prime. For illustration, the maximum for a(n) among the first 5 million terms is a(808500) = 218.

See A013916 for the above mentioned indices, numbers n such that the sum of the first n primes is prime. - M. F. Hasler, Jan 20 2014

Examples

			The sum of the first 9 primes is 100, and by adding 1 we get 101. Since 101 is a prime, a(9) = 1.
The sum of the first 10 primes is 129, since 129 - 2 = prime(31) = 127 or 129 + 2 = prime(32) = 131, a(10) = 2.
The sum of the first 129 primes minus 1 is a prime, this is 42468 - 1 = prime(4443), so a(129) = 1.
		

Links

• R. J. Cano, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007504, A101301.

Programs

PARI

a(n)=my(u=sum(j=1,n,prime(j)),k=1);while(!(isprime(u+k)||isprime(u-k)),k++);k

Formula

Algorithm:

Let S be the sum of the first n primes;

initially, let k=1;

increment k while neither S-k nor S+k is prime;

return a(n)=k.

a(n) = min(A013632(A007504(n)), A049711(A007504(n))). - M. F. Hasler, Jan 20 2014