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 A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n. 5
 2, 3, 3, 0, 0, 9 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Slow convergence. a(7) is likely either 3 or 4. Is there a simple proof that this sum converges? - Nathaniel Johnston, May 24 2011 It is apparent that the difference between this sum and the m-th partial sum (i.e., Sum_{k=2..m} (1/Sum_{j=2..k} j')) converges to C/m where C = 2.58679783..., so the convergence of the partial sums can be accelerated by a simple extrapolation (see Example section). - Jon E. Schoenfield, Jan 01 2019 LINKS EXAMPLE 1/2' + 1/(2'+3') + 1/(2'+3'+4') + 1/(2'+3'+4'+5') + ... = 1 + 1/2 + 1/6 + 1/7 + ... = 2.33009... From Jon E. Schoenfield, Jan 01 2019: (Start) Let S(m) be the m-th partial sum, i.e., S(m) = Sum_{k=2..m} (1/Sum_{j=2..k} j'), so lim_{m->inf} S(m) = A is the constant whose decimal digits are the terms of this sequence. It appears that m*(A - S(m)) approaches a constant fairly quickly, so an extrapolation such as S'(2*m) = 2*S(2*m) - S(m) allows a significant acceleration of the convergence of the partial sums. Letting f(d) = S(2^d) and evaluating f(d) at d = 0, 1, 2, ..., 29 gives the following results:                                                (f(d)-f(d-1))    d         f(d)            2*f(d) - f(d-1)       * 2^d   == ==================== ==================== =============    0 0.000000000000000000 -------------------- -------------    1 1.000000000000000000 2.000000000000000000 2.00000000000    2 1.666666666666666667 2.333333333333333333 2.66666666667    3 2.009780219780219780 2.352893772893772894 2.74490842491    4 2.168641115495430913 2.327502011210642045 2.54177433144    5 2.249470661422227179 2.330300207349023445 2.58654546966    6 2.289733935307516488 2.329997209192805797 2.57684952866    7 2.309907327553392838 2.330080719799269188 2.58219420747    8 2.319997440119766296 2.330087552686139754 2.58306881699    9 2.325043274992913392 2.330089109866060489 2.58346745505   10 2.327568044310015805 2.330092813627118217 2.58536378071   11 2.328830986923564300 2.330093929537112796 2.58650647255   12 2.329462418954684871 2.330093850985805442 2.58634559947   13 2.329778172138129318 2.330093925321573765 2.58665007878   14 2.329936056061788876 2.330093939985448435 2.58677020524   15 2.330014997129428390 2.330093938197067903 2.58674090441   16 2.330054468293665605 2.330093939457902821 2.58678221945   17 2.330074203909579733 2.330093939525493861 2.58678664910   18 2.330084071740119642 2.330093939570659550 2.58679256905   19 2.330089005663740362 2.330093939587361083 2.58679694726   20 2.330091472624831103 2.330093939585921843 2.58679619269   21 2.330092706105927748 2.330093939587024393 2.58679734879   22 2.330093322846521057 2.330093939587114365 2.58679753748   23 2.330093631216832915 2.330093939587144773 2.58679766502   24 2.330093785401996210 2.330093939587159506 2.58679778860   25 2.330093862494578594 2.330093939587160977 2.58679781329   26 2.330093901040869895 2.330093939587161197 2.58679782067   27 2.330093920314015649 2.330093939587161402 2.58679783443   28 2.330093929950588520 2.330093939587161391 2.58679783287   29 2.330093934768874959 2.330093939587161398 2.58679783490 (End) MAPLE with(numtheory); P:=proc(i) local a, b, f, n, p, pfs; a:=0; b:=0; for n from 2 to i do   pfs:=ifactors(n)[2];   f:=n*add(op(2, p)/op(1, p), p=pfs);   b:=b+f; a:=a+1/b; od; print(evalf(a, 300)); end: P(1000); MATHEMATICA digits = 5; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[Print["p(", m, ") = ", p[m]]; p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *) CROSSREFS Cf. A003415, A190144, A190145, A190147. Sequence in context: A265590 A260208 A242457 * A026932 A087401 A332915 Adjacent sequences:  A190143 A190144 A190145 * A190147 A190148 A190149 KEYWORD nonn,more,cons AUTHOR Paolo P. Lava, May 05 2011 EXTENSIONS a(6) corrected and a(7) removed by Nathaniel Johnston, May 24 2011 STATUS approved

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Last modified March 28 15:03 EDT 2020. Contains 333089 sequences. (Running on oeis4.)