A061784 Number of sums prime(i) + prime(j) that occur more than once for 1 <= i <= j <= n, where prime(k) = k-th prime.
0, 0, 0, 1, 2, 4, 7, 11, 16, 22, 27, 34, 41, 51, 61, 73, 86, 96, 110, 124, 140, 158, 175, 193, 211, 231, 252, 275, 299, 325, 348, 374, 401, 427, 456, 486, 516, 549, 581, 615, 650, 684, 722, 759, 798, 839, 879, 921, 961, 1005, 1048, 1095, 1142, 1189, 1238
Offset: 1
Keywords
Examples
Let P(n) = {2, 3, .., p_n} be the set of the first n primes. Construct S(n) = {p+q : p,q in P}. If every sum p+q were distinct, then |S(n)| would be n*(n+1)/2 = A000217(n). But in reality, for n >= 4, certain sums occur more than once. a(n) is the count of repeated values. For example, P(6) = {2, 3, 5, 7, 11, 13} yields S(6) = {4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26}, but 4 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) = 4 = A000217(n) - |S(6)|.
Programs
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Mathematica
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}]