A103598
Smallest prime p such that Sum_{primes q <= p} 1/q >= n/5.
Original entry on oeis.org
2, 2, 3, 3, 5, 11, 17, 41, 97, 277, 1013, 4789, 31627, 314723, 5195977, 159490147, 10443979657
Offset: 1
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For[n = 1, n ≤ 17, n++, sum = 0; For[k = 1, k ≤ 10^6, k++, sum = sum + 1/Prime[k]; If[sum >= n/5, Print[Prime[k]]; Break[]]]]
Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/5, m++];
Prime[m], {n, 1, 17}] (* Robert Price, Mar 27 2019 *)
A120055
a(n) is the least semiprime s such that the Sum_{semiprime i <= s} 1/i exceeds n.
Original entry on oeis.org
4, 35, 871, 43217, 5296623, 2011783478
Offset: 0
a(0)=4 because 1/4 > 0.
a(1)=35 because 1/4 + 1/6 + 1/9 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 + 1/25 + 1/26 + 1/33 + 1/34 + 1/35 = 15708817/15315300 > 1.
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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}]
A136095
a(n) is the largest k such that Sum_{i=1..k} 1/prime(i) <= n.
Original entry on oeis.org
2, 58, 361138, 43922730588128389
Offset: 1
A331930
a(n) is the smallest composite k such that Sum_{composites j = 4, ..., k} 1/j exceeds n/2.
Original entry on oeis.org
8, 16, 33, 63, 118, 216, 395, 715, 1281, 2279, 4036, 7102, 12441, 21722, 37797, 65558, 113422, 195759, 337148, 579465, 994194, 1703072, 2912869, 4975222, 8486672, 14459492, 24608418, 41837580, 71060409, 120585504, 204452804, 346372172, 586359050, 991915208
Offset: 1
a(1) = 1 because 1/4 + 1/6 = 0.41666... < 1/2 but 1/4 + 1/6 + 1/8 = 0.54166... > 1/2.
Cf.
A016088 (sum of reciprocals of primes exceeds n),
A076751 (sum of reciprocals of composites exceeds n),
A103592 (sum of reciprocals of primes exceeds n/2).
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