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a(n) is the number of arithmetic progressions prime chains in the form of p(n)-6k, p(n), p(n)+6k, while k > 0 and p(n) > 6k.
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%I #13 Apr 03 2023 10:36:12

%S 1,1,2,1,2,3,1,3,3,3,4,4,5,3,4,6,5,4,4,6,5,7,6,6,6,5,7,8,9,6,10,8,7,6,

%T 9,8,9,6,8,10,10,6,9,10,11,8,11,10,9,13,13,13,13,9,10,13,11,12,14,15,

%U 11,12,12,14,17,13,18,14,14,16,14,16,14,16,15,16

%N a(n) is the number of arithmetic progressions prime chains in the form of p(n)-6k, p(n), p(n)+6k, while k > 0 and p(n) > 6k.

%C Conjecture: a(n) > 0 for all n >= 5.

%C The Mathematica program gives term 5 through 80.

%H Definition of <a href="https://t5k.org/top20/page.php?id=14">Arithmetic Progressions of Primes</a>

%e n = 5, p(5) = 11; {5, 11, 17} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11. So a(5) = 1;

%e n = 6, p(6) = 13; {7, 13, 19} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11. So a(6) = 1;

%e ...

%e n = 10, p(10) = 29; {17, 29, 41}, {11, 29, 47}, {5, 29, 53} form Arithmetic Progressions Prime chains with difference 12, 18, 24 respectively. So a(10) = 3;

%t Table[ct = 0; p = Prime[i]; j = 0; While[j++; df = 6*j; df < p, If[(PrimeQ[p + df]) && (PrimeQ[p - df]), ct++]]; ct, {i, 5, 80}]

%Y Cf. A196934, A078498, A078497, A001748.

%K nonn,easy

%O 5,3

%A _Lei Zhou_, Oct 07 2011