%I
%S 0,0,0,1,2,4,7,11,16,22,27,34,41,51,61,73,86,96,110,124,140,158,175,
%T 193,211,231,252,275,299,325,348,374,401,427,456,486,516,549,581,615,
%U 650,684,722,759,798,839,879,921,961,1005,1048,1095,1142,1189,1238
%N Number of sums prime(i) + prime(j) that occur more than once for 1<=i<=j<=n, prime(k) = kth prime.
%e If {p+q} sums are produced by adding 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n primes gives results falling between these two extremes. E.g. S={2,3,5,7,11,13} provides 17 different sums of two, not necessarily different primes: {4,5,6,7,8,9,10,12,13,14,15,16,18,20,22,24,26}. Four sums arise more than once:10=3+7=5+5,14=3+11=7+7, 16=3+13=5+11,18=5+13=7+11. Thus a(6)=(6*7/2)17=4.
%t f[x_] := Prime[x] t1=Table[Length[Union[Flatten[Table[f[u]+f[w], {w, 1, m}, {u, 1, m}]]]], {m, 1, 75}] t=Table[(w*(w+1)/2)Part[t1, w], {w, 1, 75}]
%Y Cf. A000217, A061781.
%K nonn
%O 1,5
%A _Labos Elemer_, Jun 22 2001
