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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. 16
6, 6, 6, 12, 12, 12, 18, 18, 18, 24, 24, 24, 30, 30, 30, 36, 36, 36, 42, 42, 42, 48, 48, 48, 54, 54, 54, 60, 60, 60, 66, 66, 66, 72, 72, 72, 78, 78, 78, 84, 84, 84, 90, 90, 90, 96, 96, 96, 102, 102, 102, 108, 108, 108, 114, 114, 114, 120, 120, 120, 126, 126, 126, 132 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

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.

All terms of this sequence have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013

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

LINKS

Table of n, a(n) for n=1..64.

Kival Ngaokrajang, Illustration of initial terms

Sacred Geometry, Flower of life

FORMULA

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

a(n) = 2n + 4 - 2((n+2) mod 3). - Wesley Ivan Hurt, Jun 30 2013

a(n) = 6*A008620(n-1). - Kival Ngaokrajang, Oct 23 2015

EXAMPLE

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).

MATHEMATICA

Table[2 n + 4 - 2 Mod[n + 2, 3], {n, 66}] (* Michael De Vlieger, Oct 23 2015 *)

PROG

(PARI) vector(200, n, n--; 6*(n\3+1)) \\ Altug Alkan, Oct 23 2015

CROSSREFS

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

Sequence in context: A035019 A216057 A212096 * A315826 A304821 A315827

Adjacent sequences:  A052377 A052378 A052379 * A052381 A052382 A052383

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Mar 13 2000

STATUS

approved

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Last modified January 22 07:30 EST 2020. Contains 331139 sequences. (Running on oeis4.)