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COMMENTS
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a(n) is the k-th semiprime: k = 1, 13, 262, 10521, 1034924, 316108902; factors: 2*2, 5*7, 13*61, 23*1879, 3*1765541, 2*1005891739.
For a(1), sum = 1.025694370988488635...
for a(2), sum = 2.000920208207319042...
for a(3), sum = 3.000000294546501318...
for a(4), sum = 4.000000031685702432...
for a(5), sum = 5.000000000169814007...
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MATHEMATICA
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s = 0; k = 1; Do[ While[s <= n, If[ Plus @@ Last /@ FactorInteger@k == 2, s = N[ s + 1/k, 20]]; k++ ]; Print[{k - 1, s}]; k ++, {n, 0, 5}]
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