A016088 -id:A016088 - cp's OEIS frontend

A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

Original entry on oeis.org

1, 3, 59, 361139, 43922730588128390

Offset: 0

Eric W. Weisstein

The corresponding primes prime(a(n)) are in A016088.

Index m for which the prime harmonic number p[ m ] := Sum[ 1/Prime[ k ],{k,1,m} ] >= n.

E. Bach, D. Klyve, and J. P. Sorenson, Computing prime harmonic sums, Math. Comp. 78 (2009) 2283-2305.
Eric Weisstein's World of Mathematics, Prime Number.
Eric Weisstein's World of Mathematics, Harmonic Series of Primes.

Cf. A004080, A016088, A096232, A223037.

Cf. A024451/A002110 for Sum_{i = 1..n} 1/prime(i).

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) <= n, m++];
m, {n, 0, 4}] (* Robert Price, Mar 27 2019 *)

a(n)=my(t); forprime(p=2,, t+=1./p; if(t>n, return(primepi(p)))) \\ Charles R Greathouse IV, Apr 29 2015

From Jonathan Sondow, Apr 17 2013: (Start)
a(n) = A000720(A016088(n)) = A000720(A096232(n))+1.
a(n) = e^(e^(n + O(1))), see comment in A223037. [corrected by Charles R Greathouse IV, Aug 22 2013] (End)
a(n) = A103591(2*n). - Michel Marcus, Aug 22 2013

a(4) found by Tomás Oliveira e Silva (tos(AT)det.ua.pt), using the fourth term of A016088. - Dec 14 2005

a(0) from Jonathan Sondow, Apr 16 2013

A103591 Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/2.

Original entry on oeis.org

1, 3, 10, 59, 1413, 361139, 4833601540, 43922730588128390

Offset: 1

James R. Buddenhagen, Mar 28 2005

Cf. A016088, A046024, A103592-A103600.

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/2, m++];
m, {n, 1, 5}] (* Robert Price, Mar 27 2019 *)

a(n) = my(s = 0, k = 1); while ((s += 1/prime(k)) < n/2, k++); k \\ Michel Marcus, Aug 22 2013

a(2n) = A046024(n). - Michel Marcus, Aug 22 2013

a(7) from Robert Price, Dec 10 2013

a(8) derived from A046024 by Robert Price, Dec 10 2013

A281889 a(n) is the least integer k such that more than half of all integers are divisible by a product of n integers chosen from 2..k.

Original entry on oeis.org

3, 7, 433, 9257821

Offset: 1

Peter Munn, Feb 01 2017

The n chosen integers need not be distinct.
By "more than half of all integers" we mean more precisely "more than half of the integers in -m..m, for all sufficiently large m (depending on n)", and similarly with 1..m for "more than half of all positive integers".
Equivalently, a(n) is the least prime p such that more than half of all positive integers can be written as a product of primes of which n or more are not greater than p. (In this sense, a(n) might be called the median n-th least prime factor of the integers.)
The number of integers that satisfy the "product of primes" criterion for p = prime(m) is the same in every interval of primorial(m)^n integers and is A281891(m,n). Primorial(m) = A002110(m), product of the first m primes.
a(n) is the least k = prime(m) such that 2 * A281891(m,n) > A002110(m)^n.
a(n) is the least k such that more than half of all positive integers equate to the volume of an orthotope with integral sides at least n of which are orthogonal with length between 2 and k inclusive.
The next term is estimated to be a(5) ~ 3*10^18.

			For n=1, we have a(1) = 3 since for all m > 1, more than half of the integers in -m..m are divisible by an integer chosen from 2..3, i.e., either 2 or 3. We must have a(1) > 2, because the only integer in 2..2 is 2, but in each interval -2m-1..2m+1, only 2m+1 integers are even, so 2 is not a divisor of more than half of all integers in the precise sense given above.

Other sequences about medians of prime factors: A126282, A126283, A284411, A290154.

Cf. A002110, A014673, A016088, A027746, A115561, A281891, A309366.

A096580 a(n) = smallest m >= 2 such that Sum_{k=2..m} 1/(k*log(k)) >= n.

Original entry on oeis.org

2, 3, 28, 8718, 51426757439

Offset: 0

N. J. A. Sloane, Aug 13 2004

The sum diverges (see link), so a(n) is well-defined.

			For m = 27 the sum is 1.992912323604..., for m = 28 it is 2.0036302389..., so a(2) = 28.
For m = 8717 the sum is 2.999991290360..., for m = 8718 it is 3.0000039326..., so a(3) = 8718.

M. Goar, Olivier and Abel on series convergence: An episode from early 19th century analysis, Math. Mag., 72 (No. 5, 1999), 347-355.

Cf. A016088.

n = 0;  m = 2; sum = 1/(m*Log[m]); lst = {};
While[n <= 3,
  While[ sum < n, m++; sum += 1/(m*Log[m])];
AppendTo[lst, m];  n++]; lst (* Robert Price, Apr 09 2019 *)

Since Integral 1/(x*log(x)) dx = log log x, a(n) is close to e^(e^n) (cf. A096232, A096404, A016066).

a(n) is roughly exp(exp(n-k)), where k = 0.7946786454... - Charles R Greathouse IV, Jul 23 2007

a(3) from Robert G. Wilson v, Aug 17 2004

a(4) from Charles R Greathouse IV, Jul 23 2007

A103600 Smallest prime p such that Sum_{primes q <= p} 1/q >= n/6.

Original entry on oeis.org

2, 2, 2, 3, 3, 5, 7, 13, 29, 53, 109, 277, 809, 2741, 11789, 64663, 483281, 5195977, 85861889, 2358926351, 118185163069, 12041724518809

Offset: 1

James R. Buddenhagen, Mar 28 2005

nonn

Cf. A016088, A046024, A103591-A103599.

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/6, m++];
Prime[m], {n, 1, 17}] (* Robert Price, Mar 27 2019 *)

a(1), a(2), a(4) added for correctness, and a(19-21) included by Martin Raab, Mar 31 2009

a(22) from Charles R Greathouse IV, Jan 18 2012

A074631 a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum_{j=1..k} 1/(j-th composite), is > n.

Original entry on oeis.org

9, 44, 168, 587, 1940, 6192, 19285, 59010, 178122, 531923, 1574706, 4628338, 13521477, 39299115, 113712434, 327752962, 941457955, 2696114317, 7700146599, 21938239766

Offset: 1

Labos Elemer, Aug 27 2002

			1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1045/1008, but if 1/16 is not present, the sum is less than 1; 16 is the ninth composite number, so a(1) = 9.

Cf. A002387, A002808, A004080, A016088, A046024, A065855, A076751.

NextComposite[n_] := Block[{k = n + 1}, While[PrimeQ[k], k++ ]; k]; s=0; k = 4; Do[While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k - PrimePi[k] - 1]; k = NextComposite[k], {n, 1, 20}]
Table[Position[Accumulate[1/Select[Range[5*10^6],CompositeQ]],?(#>n&),1,1],{n,12}]//Flatten (* The program generates the first 12 terms of the sequence. *) (* _Harvey P. Dale, Jan 22 2023 *)

lista(cmax) = {my(n = 1, s = 0, k = 0); forcomposite(c = 1, cmax, k++; s += 1/c; if(s > n, print1(k, ", "); n++));} \\ Amiram Eldar, Jul 17 2024

a(n) = Min { k : Sum_{j=1..k} 1/A002808(j) > n }.
Limit_{n->oo} a(n+1)/a(n) = e. - Robert G. Wilson v, Aug 28 2002
a(n) = A065855(A076751(n)). - Amiram Eldar, Jul 17 2024

Edited by Robert G. Wilson v, Aug 28 2002
More terms from Robert Gerbicz, Aug 30 2002
2 more terms from Robert G. Wilson v, Sep 03 2002
Edited by Jon E. Schoenfield, Sep 13 2023
a(18)-a(20) from Amiram Eldar, Jul 17 2024

A076751 a(n) is the smallest composite k such that Sum_{composites j = 4, ..., k} 1/j exceeds n.

Original entry on oeis.org

16, 63, 216, 715, 2279, 7102, 21722, 65558, 195759, 579465, 1703072, 4975222, 14459492, 41837580, 120585504, 346372172, 991915208, 2832896772, 8071045528, 22944211170

Offset: 1

Jack Brennen, Nov 12 2002

These partial sums, like the harmonic sequence (A004080), can never be integers.

			a(1) = 1 because 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 = 0.97420... < 1 but 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1.03670... > 1.

Cf. A002808, A004080, A016088, A074631.

NextComposite[n_] := Block[{k = n + 1}, While[ PrimeQ[k], k++ ]; k]; k = 4; s = 0; Do[ While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k]; k = NextComposite[k], {n, 1, 17}]

lista(cmax) = {my(n = 1, s = 0); forcomposite(c = 1, cmax, s += 1/c; if(s > n, print1(c, ", "); n++));} \\ Amiram Eldar, Jul 17 2024

Limit_{n->oo} a(n+1)/a(n) = e.

a(n) = A002808(A074631(n)). - Amiram Eldar, Jul 17 2024

Edited and extended by Robert G. Wilson v, Nov 14 2002
Name edited and a(18) added by Jon E. Schoenfield, Feb 01 2020
a(19)-a(20) from Amiram Eldar, Jul 17 2024

A223037 a(n) = largest prime p such that Sum_{primes q = 2, ..., p} 1/q does not exceed n.

Original entry on oeis.org

3, 271, 5195969, 1801241230056600467

Offset: 1

Jonathan Sondow, Apr 16 2013

Since Sum_{prime q} 1/q diverges, the sequence is infinite.
In fact, by the Prime Number Theorem Prime(k) ~ k log k as k -> infinity, and by integration Sum_{k <= n} 1/(k log k) ~ log log n, so a(n) ~ Prime(Floor(e^e^n)).
a(4) = A000040(A046024(4)-1) = Prime[43922730588128389], but Mathematica 7.0.0 cannot compute this prime on a Mac computer running OS X.
Instead, using a(4) = largest prime < A016088(4) = 1801241230056600523, Mathematica's PrimeQ function finds that a(4) = 1801241230056600467.
See A016088 for other relevant comments, references, links, and programs.

			a(1) = 3 because 1/2 + 1/3 < 1 < 1/2 + 1/3 + 1/5.

Cf. A016088, A046024, A024451, A096232.

a(n) = A000040(A046024(n)-1) = largest prime < A016088(n).

a(n) ~ Prime(Floor(e^e^n)) = A000040(A096232(n)) as n -> infinity.

A103592 a(n) is the smallest prime p such that Sum_{primes q <= p} 1/q >= n/2.

Original entry on oeis.org

2, 5, 29, 277, 11789, 5195977, 118185163069, 1801241230056600523

Offset: 1

James R. Buddenhagen, Mar 28 2005

nonn

Cf. A016088, A046024, A103591-A103600.

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) < n/2, m++];
Prime[m], {n, 1, 8}] (* Robert Price, Mar 27 2019 *)

a(7)-a(8) from Martin Raab, Aug 24 2008

a(7) corrected by Martin Raab, Mar 31 2009

A119494 a(n) = smallest prime number p_k such that 1/p_n + 1/p_{n+1} + ... + 1/p_k > 1.

Original entry on oeis.org

5, 29, 109, 347, 857, 1627, 2999, 4931, 7759, 11677, 16111, 22229, 29269, 37717, 48527, 61057, 75503, 91463, 110567, 131671, 155509, 183587, 214189, 248597, 286073, 325889, 369983, 419459, 473659, 534043, 600631, 667547, 739549, 816779, 901007, 988661

Offset: 1

Charles R Greathouse IV, May 25 2006

nonn

Domaratzki, Ellul, Shallit, & Wang call the n-th term of A092325 ϖ(n), and A092325(n) = pi(a(n)). - Charles R Greathouse IV, Aug 08 2016

			a(2) = 29 because 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 + 1/29 = 1.0334... > 1 and 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 = 0.9989... < 1.

J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 76, entry 347 and page 108, entry 857.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Michael Domaratzki, Keith Ellul, Jeffrey Shallit and Ming-Wei Wang, Non-Uniqueness and Radius of Cyclic Unary NFAs, International Journal of Foundations of Computer Science, Vol. 16, No. 5 (2005) pp. 883-896, alternative link.

Cf. A092325, A016088.

f[0]={0,0}; f[n_] := f[n] = Module[{f1=f[n-1]}, p=f1[[1]]; s=f1[[2]]-If[n>1, 1/Prime[n-1], 0]; While[s<1, p=NextPrime[p]; s+=1/p]; {p,s}]; f[#][[1]] & /@ Range[30] (* Amiram Eldar, Dec 24 2018 *)

a(n)=my(s=0.);forprime(p=prime(n),default(primelimit),s+=1/p;if(s>1,return(p)))

a(n) is approximately prime(n)^e.

a(n) = prime(A092325(n)). - Amiram Eldar, Dec 24 2018

Definition corrected by Ray Chandler, Jun 09 2006
Edited by Charles R Greathouse IV, Nov 12 2009
a(35)-a(36) from Amiram Eldar, Dec 24 2018

cp's OEIS Frontend

A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

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A103591 Smallest number m such that Sum_{k=1..m} 1/prime(k) >= n/2.

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A281889 a(n) is the least integer k such that more than half of all integers are divisible by a product of n integers chosen from 2..k.

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A096580 a(n) = smallest m >= 2 such that Sum_{k=2..m} 1/(k*log(k)) >= n.

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A103600 Smallest prime p such that Sum_{primes q <= p} 1/q >= n/6.

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A074631 a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum_{j=1..k} 1/(j-th composite), is > n.

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A076751 a(n) is the smallest composite k such that Sum_{composites j = 4, ..., k} 1/j exceeds n.

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A223037 a(n) = largest prime p such that Sum_{primes q = 2, ..., p} 1/q does not exceed n.

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