A240087 - OEIS

%I #18 Feb 28 2023 09:14:25

%S 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,

%T 48,30,84,18,12,66,60,84,24,6,36,18,6,54,84,48,36,18,36,12,126,54,6,

%U 42,18,54,36,6,12,48,24,6,30,36,24,108,90,36,18,42,66,36,6

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

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

%H Lei Zhou, <a href="/A240087/b240087.txt">Table of n, a(n) for n = 2..10001</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeArithmeticProgression.html">Prime Arithmetic Progression</a>.

%F 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

%e 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.

%e 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.

%e 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.

%t 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}]

%Y Cf. A000040.

%K nonn

%O 2,1

%A _Lei Zhou_, Mar 31 2014

%E Name and examples edited, link added. - _Wolfdieter Lang_, Apr 17 2014