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%I #6 Jan 20 2021 08:45:17
%S 1,3,5,9,14,20,25,32,39,49,56,68,78,92,105,115,128,144,157,171,192,
%T 211,231,253,276,297,319,339,366,396,419,442,473,500,533,561,592,628,
%U 665,691,726,759,794,832,868,900,936,979,1028,1070,1114,1159,1208,1248,1298
%N Number of distinct integers expressible as sums of consecutive primes up to n-th prime.
%C 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.
%H Zak Seidov, <a href="/A166709/b166709.txt">Table of n, a(n) for n=1..1000</a>
%e 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);
%e 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.
%e From _Rick L. Shepherd_, Oct 18 2009: (Start)
%e The first 6 rows of actual sums are:
%e n=1: 2
%e n=2: 2,3,5
%e n=3: 2,3,5,8,10
%e n=4: 2,3,5,7,8,10,12,15,17
%e n=5: 2,3,5,7,8,10,11,12,15,17,18,23,26,28
%e n=6: 2,3,5,7,8,10,11,12,13,15,17,18,23,24,26,28,31,36,39,41 (End)
%t Table[Length[Union[Total/@Flatten[Table[Partition[Prime[Range[m]],k,1],{k,m}],1]]],{m,100}]
%o (PARI) A166709(n)=#Set(concat(vector(n,i,vector(i,j,sum(k=j,i,prime(k)))))) \\ _M. F. Hasler_, Oct 18 2009
%Y Cf. A034707 (numbers which are sums of consecutive primes).
%Y Cf. A152415, A152430.
%K nonn
%O 1,2
%A _Zak Seidov_, Oct 18 2009
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