A103599
Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/6.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 6, 10, 16, 29, 59, 140, 400, 1413, 6467, 40261, 361139, 4990906, 114916199, 4833601540
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/6, Print[k]; Break[]]]] (* Robert Price, Dec 09 2013 *)
Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/6, m++];
m, {n, 1, 17}] (* Robert Price, Mar 27 2019 *)
a(2)-a(4) and a(19)-a(21) added by
Robert Price, Dec 10 2013
A332991
a(n) is the smallest prime power p^k (p prime, k >= 1) such that sum of reciprocals of prime powers up to p^k (prime power harmonic sum) exceeds n.
Original entry on oeis.org
2, 4, 19, 1307, 266655247
Offset: 0
a(1) = 4 because 1/2 + 1/3 = 0.8333... < 1 but 1/2 + 1/3 + 1/4 = 1.0833... > 1.
Similar sequences:
A002387 (for positive integers),
A016088 (for primes),
A076751 (for composite numbers),
A333004 (for squarefree numbers).
A333004
a(n) is the smallest squarefree number k such that sum of reciprocals of squarefree numbers up to k (squarefree harmonic sum) exceeds n.
Original entry on oeis.org
1, 2, 5, 26, 130, 670, 3466, 17985, 93179, 482762, 2501013, 12956855, 67125243, 347753857, 1801596939
Offset: 0
a(2) = 5 because 1/1 + 1/2 + 1/3 = 1.8333... < 2 but 1/1 + 1/2 + 1/3 + 1/5 = 2.0333... > 2.
A347310
a(n) = smallest k such that Sum_{i=1..k} log(p_i)/p_i >= n, where p_i is the i-th prime.
Original entry on oeis.org
3, 8, 19, 43, 100, 236, 562, 1354, 3300, 8119, 20136, 50302, 126451, 319628, 811829, 2070790, 5302162, 13621745, 35101258, 90696900, 234924747, 609864582, 1586430423, 4134442382, 10793331294, 28221407514, 73898377351
Offset: 1
a(1) = 3 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Society, 2015. See page 16.
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Table[k=1;While[Sum[Log@Prime@i/Prime@i,{i,++k}]Giorgos Kalogeropoulos, Sep 08 2021 *)
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a(n) = my(k=0, s=0, p=2); while (s < n, s += log(p)/p; k++; p = nextprime(p+1)); k; \\ Michel Marcus, Sep 06 2021
A347311
a(n) is the smallest prime q such that Sum_{primes p <= q} log(p)/p >= n.
Original entry on oeis.org
5, 19, 67, 191, 541, 1487, 4079, 11173, 30559, 83137, 226427, 615919, 1675771, 4556771, 12387481, 33677717, 91558231, 248887319, 676566619, 1839125531, 4999337929, 13589640521, 36940536917, 100415101481, 272957090657, 741974865617, 2016896970001
Offset: 1
a(1) = 5 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Society, 2015. See page 16.
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Table[i=1;d=Log@Prime@i/Prime@i;While[dGiorgos Kalogeropoulos, Sep 08 2021 *)
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a(n) = my(k=0, s=0, p=2); while (s < n, s += log(p)/p; k++; p = nextprime(p+1)); prime(k); \\ Michel Marcus, Sep 06 2021
A103593
Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/3.
Original entry on oeis.org
1, 2, 3, 6, 16, 59, 400, 6467, 361139, 114916199
Offset: 1
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Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/3, m++];
m, {n, 1, 10}] (* Robert Price, Mar 27 2019 *)
A103594
Smallest prime p such that Sum_{primes q <= p} 1/q >= n/3.
Original entry on oeis.org
2, 3, 5, 13, 53, 277, 2741, 64663, 5195977, 2358926351, 12041724518809, 1801241230056600523
Offset: 1
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Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/3, m++];
Prime[m], {n, 1, 10}] (* Robert Price, Mar 27 2019 *)
A103595
Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/4.
Original entry on oeis.org
1, 1, 2, 3, 5, 10, 21, 59, 231, 1413, 15474, 361139, 22347214, 4833601540
Offset: 1
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Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/4, m++];
m, {n, 1, 14}] (* Robert Price, Mar 27 2019 *)
A103596
Smallest prime p such that Sum_{primes q <= p} 1/q >= n/4.
Original entry on oeis.org
2, 2, 3, 5, 11, 29, 73, 277, 1453, 11789, 169751, 5195977, 420055319, 118185163069
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/4, Print[Prime[k]]; Break[]]]]
Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/4, m++];
Prime[m], {n, 1, 14}] (* Robert Price, Mar 27 2019 *)
A103597
Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/5.
Original entry on oeis.org
1, 1, 2, 2, 3, 5, 7, 13, 25, 59, 170, 644, 3402, 27178, 361139, 8947437, 474314304
Offset: 1
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Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/5, m++];
m, {n, 1, 17}] (* Robert Price, Mar 27 2019 *)
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