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%I #18 Sep 13 2023 08:28:56
%S 9,44,168,587,1940,6192,19285,59010,178122,531923,1574706,4628338,
%T 13521477,39299115,113712434,327752962,941457955
%N a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum 1/(j-th composite), is > n.
%C Limit_{n->oo} a(n+1)/a(n) = e. - _Robert G. Wilson v_, Aug 28 2002
%F a(n) = Min { k : Sum_{j=1..k} 1/A002808(j) > n }.
%e 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.
%t 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}]
%t 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 *)
%Y Cf. A002387, A016088, A046024, A002808, A004080.
%K nonn,nice,more
%O 1,1
%A _Labos Elemer_, Aug 27 2002
%E Edited by _Robert G. Wilson v_, Aug 28 2002
%E More terms from _Robert Gerbicz_, Aug 30 2002
%E 2 more terms from _Robert G. Wilson v_, Sep 03 2002
%E Edited by _Jon E. Schoenfield_, Sep 13 2023
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