A240087 Smallest difference to start a prime arithmetic progression of three or more terms with the n-th prime number.

2, 6, 6, 18, 24, 36, 24, 24, 30, 6, 36, 6, 18, 12, 18, 24, 6, 42, 78, 78, 24, 48, 12, 6, 6, 48, 30, 84, 18, 12, 66, 60, 84, 24, 6, 36, 18, 6, 54, 84, 48, 36, 18, 36, 12, 126, 54, 6, 42, 18, 54, 36, 6, 12, 48, 24, 6, 30, 36, 24, 108, 90, 36, 18, 42, 66, 36, 6

Offset: 2

Author

Lei Zhou, Mar 31 2014

Keywords

• nonn

Comments

It is conjectured that this sequence is defined for all odd prime numbers.

Examples

			n=2: the second prime number is 3; 3, 5, 7 form a 3-term prime arithmetic progression with difference 2. So a(2) = 2.
n=3: the third prime is 5; 5, 11, 17, 23, 29 form a 5-term prime arithmetic progression with difference 6, and this is the smallest difference to obtain three or more terms, hence a(3) = 6.
n=5: the fifth prime number is 11. Although 11, 17, 23, 29 form a 4-term prime arithmetic progression with difference 6, this prime arithmetic progression actually starts with 5 (see n=3). 11, 29, 47 form a 3-term prime arithmetic progression with difference 18. So a(5) = 18.
		

Links

• Lei Zhou, Table of n, a(n) for n = 2..10001
• Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.

Crossrefs

Cf. A000040.

Programs

Mathematica

Table[p = Prime[n]; pt = p; While[pt = NextPrime[pt]; diff = pt - p; ! ((PrimeQ[pt + diff]) && ((! (PrimeQ[p - diff])) || (p < diff)))]; diff, {n, 2, 69}]

Formula

prime(n) + a(n)*k, with n >= 2, for k = 0, 1, 2, ..., kmax(n), with kmax(n) >= 2, are primes, but prime(n) - a(n) is not a prime. prime(n)= A000040(n). - Wolfdieter Lang, Apr 17 2014

Extensions

Name and examples edited, link added. - Wolfdieter Lang, Apr 17 2014