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Problem 196

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Prime triplets

Build a triangle from all positive integers in the following way:

1
2	3
4	5	6
7	8	9	10
11	12	13	14	15
16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35	36
37	38	39	40	41	42	43	44	45
46	47	48	49	50	51	52	53	54	55
56	57	58	59	60	61	62	63	64	65	66
..	..	..

Each positive integer has up to eight neighbours in the triangle.

A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.

For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.

If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31.
If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.

Define S(n) as the sum of the primes in row n which are elements of any prime triplet.
Then S(8)=60 and S(9)=37.

You are given that S(10000)=950007619.

Find  S(5678027) + S(7208785).

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素数三元组

将所有正整数按如下方式排列成三角形的样式：

1
2	3
4	5	6
7	8	9	10
11	12	13	14	15
16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35	36
37	38	39	40	41	42	43	44	45
46	47	48	49	50	51	52	53	54	55
56	57	58	59	60	61	62	63	64	65	66
..	..	..

每个正整数在三角形中有最多八个邻居。

如果三个素数中，有两个均为另一个的邻居，那么这一组三个素数被称为一个素数三元组。

例如，在第二行，素数2和3都是一些素数三元组的元素。

如果我们看第8行，将有两个素数属于某些素数三元组，这两个素数是29和31。
如果我们看第9行，只有一个素数属于某些素数三元组，这个素数是37。

记S(n)是第n行中属于某些素数三元组的素数之和。
因此S(8)=60，而S(9)=37。

已知S(10000)=950007619。

求S(5678027) + S(7208785)。

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