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A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.

%I A240233 #20 Dec 07 2015 01:03:26
%S A240233 5,5,5,5,7,7,5,5,5,7,5,7,11,5,11,5,7,23,13,11,5,5,41,5,7,37,29,11,5,
%T A240233 13,7,5,13,23,13,7,5,5,23,11,11,5,5,13,7,5,29,23,13,7,5,19,41,13,17,
%U A240233 11,7,5,19,7,7,7,5,5,7,5,7,11,29,13,5,17,5,19,7,7,5
%N A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.
%C A240233 a(n), a(n) + 6n, and a(n) + 12n form an arithmetic progression with a common difference of 6n.
%C A240233 If the interval is not a multiple of six, such an arithmetic progression of primes cannot exist unless a(n)=3. For example, 3,5,7 has an interval of 2; 3,7,11 has an interval of 4; and 3,11,19 has an interval of 8, as in A115334 and A206037.
%C A240233 Conjecture: a(n) is defined for all n > 0.
%H A240233 Lei Zhou, <a href="/A240233/b240233.txt">Table of n, a(n) for n = 1..10000</a>
%e A240233 n=1, 6n=6. 5,11,17 are all prime numbers with an interval of 6. So a(1)=5;
%e A240233 ...
%e A240233 n=13, 6n=78. 5+78=83, 5+2*78=161=7*23(x); 7+78=85(x); 11+78=89, 11+78*2=167. 11,89,167 are all prime numbers with an interval of 78. So a(13)=11.
%t A240233 Table[diff = n*6; k = 1; While[k++; p = Prime[k]; cp1 = p + diff; cp2 = p + 2*diff; ! ((PrimeQ[cp1]) && (PrimeQ[cp2]))]; p, {n, 77}]
%Y A240233 Cf. A000040, A115334, A206037, A240087.
%K A240233 nonn,easy
%O A240233 1,1
%A A240233 _Lei Zhou_, Apr 02 2014