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A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.

%I A052380 #39 Apr 25 2018 03:19:12
%S A052380 6,6,6,12,12,12,18,18,18,24,24,24,30,30,30,36,36,36,42,42,42,48,48,48,
%T A052380 54,54,54,60,60,60,66,66,66,72,72,72,78,78,78,84,84,84,90,90,90,96,96,
%U A052380 96,102,102,102,108,108,108,114,114,114,120,120,120,126,126,126,132
%N A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.
%C A052380 For d=D the quadruple of primes becomes a triple: [p,p+d],[p+d,p+2d].
%C A052380 Without the p > 3 condition, a(1)=2.
%C A052380 The starter prime p, is followed by a prime d-pattern of [d,D-d,d], where D-d=a(n)-2n is 4,2 or 0; these d-patterns are as follows: [2,4,2], [4,2,4], [6,6], [8,4,8], [10,2,10], [12,12], etc.
%C A052380 All terms of this sequence have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013
%C A052380 a(n+1) is also the number of the circles added at the n-th iteration of the pattern generated by the construction rules: (i) At n = 0, there are six circles of radius s with centers at the vertices of a regular hexagon of side length s. (ii) At n > 0, draw a circle with center at each boundary intersection point of the figure of the previous iteration. The pattern seems to be the flower of life except at the central area. See illustration. - _Kival Ngaokrajang_, Oct 23 2015
%H A052380 Kival Ngaokrajang, <a href="/A052380/a052380.pdf">Illustration of initial terms</a>
%H A052380 Sacred Geometry, <a href="http://www.bibliotecapleyades.net/geometria_sagrada/esp_geometria_sagrada_6.htm">Flower of life</a>
%F A052380 a(n) = 6*ceiling(n/3) = 6*ceiling(d/6) = D = D(n).
%F A052380 a(n) = 2n + 4 - 2((n+2) mod 3). - _Wesley Ivan Hurt_, Jun 30 2013
%F A052380 a(n) = 6*A008620(n-1). - _Kival Ngaokrajang_, Oct 23 2015
%e A052380 n=5, d=2n=10, the minimal distance for 10-twins is 12 (see A031928, d=10) the smallest term in A053323. It occurs first between twins of [409,419] and [421,431]; see 409 = A052354(1) = A052376(1) = A052381(5).
%t A052380 Table[2 n + 4 - 2 Mod[n + 2, 3], {n, 66}] (* _Michael De Vlieger_, Oct 23 2015 *)
%o A052380 (PARI) vector(200, n, n--; 6*(n\3+1)) \\ _Altug Alkan_, Oct 23 2015
%Y A052380 Cf. A001223, A031924-A031938, A053319-A053331, A052350-A052358, A008620.
%K A052380 nonn,easy
%O A052380 1,1
%A A052380 _Labos Elemer_, Mar 13 2000