For n=4; 7 is the 4th prime. 7 = 7, 9 = 2+7, 10 = 3+7, 12 = 5+7 = 2+3+7, 14 = 2+5+7, 15 = 3+5+7, 17 = 2+3+5+7. Values of m are 7 and 9,10,12,14,15,17. so a(4)=7.
7 = 7, so 7*2 = 14 = 24-10 = 24+(-2-3-5).
2+7 = 9, so (2+7)*2 = 18 = 24- 6 = 24+( 2-3-5).
3+7 = 10, so (3+7)*2 = 20 = 24- 4 = 24+(-2+3-5).
5+7 = 12, so (5+7)*2 = 24 = 24+ 0 = 24+(-2-3+5).
2+5+7 = 14, so (2+5+7)*2 = 28 = 24+ 4 = 24+( 2-3+5).
3+5+7 = 15, so (3+5+7)*2 = 30 = 24+ 6 = 24+(-2+3+5).
2+3+5+7 = 17. so (2+3+5+7)*2 = 34 = 24+10 = 24+( 2+3+5). (End)
Let b(n) be the number of k (>=0) that can be expressed as the sum of distinct primes with largest prime in the sum not greater than prime(n).
n |b(n)| |
--+----+------------+--------------------------------------
4 | 12 | 0 | 11
| | 2 | 13 = 2+11
| | 3 | 14 = 3+11
| | 5 | 16 = 5+11
| | 7 | 18 = 7+11
| | 8 = 3+5 | 19 = 8+11 = (3+5)+11
| | 9 = 17-8 | 20 = 9+11 = (2+3+5+7)-(3+5)+11
| | 10 = 17-7 | 21 = 10+11 = (2+3+5+7)-7 +11
| | 12 = 17-5 | 23 = 12+11 = (2+3+5+7)-5 +11
| | 14 = 17-3 | 25 = 14+11 = (2+3+5+7)-3 +11
| | 15 = 17-2 | 26 = 15+11 = (2+3+5+7)-2 +11
| | 17 = 17-0 | 28 = 17+11 = (2+3+5+7) +11
--+----+------------+--------------------------------------
5 | 23 | 0 | 13
| | 2 | 15 = 2+13
| | 3 | 16 = 3+13
| | 5 | 18 = 5+13
| | 7 | 20 = 7+13
| | 8 = 3+5 | 21 = 8+13 = (3+5) +13
| | 9 = 2+7 | 22 = 9+13 = (2+7) +13
| | 10 = 2+3+5 | 23 = 10+13 = (2+3+5)+13
| | 11 | 24 = 11+13
| | ... | ...
| | 17 = 28-11 | 30 = 17+13 = (2+3+5+7+11)-11 +13
| | 18 = 28-10 | 31 = 18+13 = (2+3+5+7+11)-(2+3+5)+13
| | 19 = 28- 9 | 32 = 19+13 = (2+3+5+7+11)-(2+7) +13
| | 20 = 28- 8 | 33 = 20+13 = (2+3+5+7+11)-(3+5) +13
| | 21 = 28- 7 | 34 = 21+13 = (2+3+5+7+11)- 7 +13
| | 23 = 28- 5 | 36 = 23+13 = (2+3+5+7+11)- 5 +13
| | 25 = 28- 3 | 38 = 25+13 = (2+3+5+7+11)- 3 +13
| | 26 = 28- 2 | 39 = 26+13 = (2+3+5+7+11)- 2 +13
| | 28 = 28- 0 | 41 = 28+13 = (2+3+5+7+11) +13
--+----+------------+-------------------------------------
...
b(n) = Sum_{k=1..n} prime(k) + 1 - 3*2 = A007504(n) - 5 for n>3.
So a(n) = b(n-1) = A007504(n-1) - 5 for n>4.
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