A052380 - OEIS

%I #39 Apr 25 2018 03:19:12

%S 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 54,54,54,60,60,60,66,66,66,72,72,72,78,78,78,84,84,84,90,90,90,96,96,

%U 96,102,102,102,108,108,108,114,114,114,120,120,120,126,126,126,132

%N 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 For d=D the quadruple of primes becomes a triple: [p,p+d],[p+d,p+2d].

%C Without the p > 3 condition, a(1)=2.

%C 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 All terms of this sequence have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013

%C 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 Kival Ngaokrajang, <a href="/A052380/a052380.pdf">Illustration of initial terms</a>

%H Sacred Geometry, <a href="http://www.bibliotecapleyades.net/geometria_sagrada/esp_geometria_sagrada_6.htm">Flower of life</a>

%F a(n) = 6*ceiling(n/3) = 6*ceiling(d/6) = D = D(n).

%F a(n) = 2n + 4 - 2((n+2) mod 3). - _Wesley Ivan Hurt_, Jun 30 2013

%F a(n) = 6*A008620(n-1). - _Kival Ngaokrajang_, Oct 23 2015

%e 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 Table[2 n + 4 - 2 Mod[n + 2, 3], {n, 66}] (* _Michael De Vlieger_, Oct 23 2015 *)

%o (PARI) vector(200, n, n--; 6*(n\3+1)) \\ _Altug Alkan_, Oct 23 2015

%Y Cf. A001223, A031924-A031938, A053319-A053331, A052350-A052358, A008620.

%K nonn,easy

%O 1,1

%A _Labos Elemer_, Mar 13 2000