A166709 Number of distinct integers expressible as sums of consecutive primes up to n-th prime.

1, 3, 5, 9, 14, 20, 25, 32, 39, 49, 56, 68, 78, 92, 105, 115, 128, 144, 157, 171, 192, 211, 231, 253, 276, 297, 319, 339, 366, 396, 419, 442, 473, 500, 533, 561, 592, 628, 665, 691, 726, 759, 794, 832, 868, 900, 936, 979, 1028, 1070, 1114, 1159, 1208, 1248, 1298

Offset: 1

Author

Zak Seidov, Oct 18 2009

Keywords

• nonn

Comments

a(n) <= n(n+1)/2 (= T(n), A000217 Triangular numbers) because some sums give the same value. E.g., a(4)=9, T(4)=10, a(4)=T(4)-1, because 5 is equal to two sums 2+3, and 5. For n=100 "deficit" is 700: a(100)=4350=T(100)-700=5050-700.

Examples

			n=4: 9 distinct integers = 2, 3, 5, 7, 8(=3+5), 10(=2+3+5), 12(=5+7), 15(=3+5+7), and 17(=2+3+5+7);
n=10: 49 distinct integers: 2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 23, 24, 26, 28, 29, 30, 31, 36, 39, 41, 42, 48, 49, 52, 53, 56, 58, 59, 60, 67, 71, 72, 75, 77, 83, 88, 90, 95, 98, 100, 101, 112, 119, 124, 127, 129.
From _Rick L. Shepherd_, Oct 18 2009: (Start)
The first 6 rows of actual sums are:
n=1: 2
n=2: 2,3,5
n=3: 2,3,5,8,10
n=4: 2,3,5,7,8,10,12,15,17
n=5: 2,3,5,7,8,10,11,12,15,17,18,23,26,28
n=6: 2,3,5,7,8,10,11,12,13,15,17,18,23,24,26,28,31,36,39,41 (End)
		

Links

• Zak Seidov, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A034707 (numbers which are sums of consecutive primes).

Cf. A152415, A152430.

Programs

Mathematica

Table[Length[Union[Total/@Flatten[Table[Partition[Prime[Range[m]],k,1],{k,m}],1]]],{m,100}]

PARI

A166709(n)=#Set(concat(vector(n,i,vector(i,j,sum(k=j,i,prime(k)))))) \\ M. F. Hasler, Oct 18 2009