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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

Table of n, a(n) for n=1..6.

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.)