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.
9, 44, 168, 587, 1940, 6192, 19285, 59010, 178122, 531923, 1574706, 4628338, 13521477, 39299115, 113712434, 327752962, 941457955, 2696114317, 7700146599, 21938239766
Offset: 1
Examples
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.
Programs
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Mathematica
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 *) -
PARI
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
Formula
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
Extensions
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