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

2,1

Comments

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

Links

Lei Zhou, Table of n, a(n) for n = 2..10001

Eric Weisstein's World of Mathematics, Prime Arithmetic Progression.

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

Example

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.

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}]

Crossrefs

Cf. A000040.

Sequence in context: A019076 A197109 A111410 * A083774 A081518 A367767

Adjacent sequences: A240084 A240085 A240086 * A240088 A240089 A240090

Keyword

nonn

Author

Lei Zhou, Mar 31 2014

Extensions

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

Status

approved