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

%I A240087 #19 Feb 16 2025 08:33:21
%S A240087 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 A240087 48,30,84,18,12,66,60,84,24,6,36,18,6,54,84,48,36,18,36,12,126,54,6,
%U A240087 42,18,54,36,6,12,48,24,6,30,36,24,108,90,36,18,42,66,36,6
%N A240087 Smallest difference to start a prime arithmetic progression of three or more terms with the n-th prime number.
%C A240087 It is conjectured that this sequence is defined for all odd prime numbers.
%H A240087 Lei Zhou, <a href="/A240087/b240087.txt">Table of n, a(n) for n = 2..10001</a>
%H A240087 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeArithmeticProgression.html">Prime Arithmetic Progression</a>.
%F A240087 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 A240087 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 A240087 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 A240087 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 A240087 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 A240087 Cf. A000040.
%K A240087 nonn
%O A240087 2,1
%A A240087 _Lei Zhou_, Mar 31 2014
%E A240087 Name and examples edited, link added. - _Wolfdieter Lang_, Apr 17 2014