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Number of prime triples (p,q,r) <= n with p<q<r=p+6.
4

%I #11 Feb 16 2025 08:32:54

%S 0,0,0,0,1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

%T 5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,

%U 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,10,10,11

%N Number of prime triples (p,q,r) <= n with p<q<r=p+6.

%C Convention: a prime triple is <= n iff its smallest member is <= n;

%C a(n) <= A098428(n).

%H Harvey P. Dale, <a href="/A098424/b098424.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>

%e a(15) = #{(5,7,11),(7,11,13),(11,13,17),(13,17,19)} = 4.

%t With[{pts=Select[Partition[Prime[Range[1200]],3,1],Last[#]-First[#] == 6&]}, Table[Count[pts,_?(First[#]<=n&)],{n,110}]] (* _Harvey P. Dale_, Nov 09 2011 *)

%o (Haskell)

%o a098424 n = length [(p,q,r) | p <- takeWhile (<= n) a000040_list,

%o let r = p + 6, a010051 r == 1, q <- [p+1..r-1], a010051 q == 1]

%o -- _Reinhard Zumkeller_, Nov 15 2011

%Y Cf. A007529, A098414, A098415, A071538.

%Y Cf. A010051, A000040.

%K nonn,changed

%O 1,7

%A _Reinhard Zumkeller_, Sep 07 2004