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A278389 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(k*prime(k)). 0
3, 7, 4, 4, 8, 5, 1, 8, 7, 9, 7, 4, 7, 4, 6, 1, 6, 3, 2, 1, 7, 0, 9, 4, 0, 8, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Alternating sum of the reciprocals of the products of k and the k-th prime.

From Jon E. Schoenfield, Jan 15 2021: (Start)

The second Mathematica program appears to compute partial sums through k=10^10 and includes a comment that it is "good for the first 27 digits". For the (10^10-1)st, (10^10)th and (10^10+1)st partial sums, I get

.

     k        prime(k)         S(k) = k-th partial sum

=========== ============ ====================================

.9999999999 252097800611 0.3744851879747461632172924014592...

10000000000 252097800623 0.3744851879747461632168957300096...

10000000001 252097800629 0.3744851879747461632172924014591...

.

Since the sum is alternating, successive partial sums oscillate around a central curve, and taking the mean of the k-th and (k+1)st partial sums gives an estimate of the value around which the sum appears to be converging. Those means for k = 9999999999 and 10000000000 are

.

           k                (S(k) + S(k+1))/2

      =========== =====================================

       9999999999 0.37448518797474616321709406573443...

      10000000000 0.37448518797474616321709406573440...

.

Of course, even though these results happen to agree through their first 31 significant digits, they are certainly nowhere near sufficient to establish the first 31 significant digits of the infinite sum (since the sequence of mean partial sums tends to meander). However, these results are clearly lower than the value given in the Data (i.e., 0.374485187974746163217094086), so (if correct) they would appear to indicate either that the last two terms in the Data are incorrect or that those terms were obtained using partial sums significantly beyond the (10^10)th. (End)

LINKS

Table of n, a(n) for n=0..26.

Eric Weisstein's World of Mathematics, Prime Sums

FORMULA

Sum_{k>=1} (-1)^(k+1)/(k*prime(k)) = 1/(1*2) - 1/(2*3) + 1/(3*5) - 1/(4*7) + 1/(5*11) - ... .

EXAMPLE

0.374485187974746163217094086...

MATHEMATICA

RealDigits[N[-Sum[(-1)^k/(k*Prime[k]), {k, 1, 8*10^6}], 30]][[1]] (* G. C. Greubel, Nov 22 2016 *)

s = 0; k = 1; p = 2; While[k =< 10^10, s = N[s - (-1)^k/(k*p), 48]; k++; p = NextPrime@p]; RealDigits[s, 10, 20] (* good for the first 27 digits *) (* Robert G. Wilson v, Mar 07 2019 *)

CROSSREFS

See the following for alternating sums of reciprocals of primes, composites, and related expressions: A078437 (primes), A242301 (primes^2), A242302 (primes^3), A242303 (primes^4), A242304 (primes^5), A269229 (composites), A275110 (composites excluding prime powers), A275712 (nonprimes), A276494 (composites^2).

Sequence in context: A256676 A267412 A087941 * A021271 A238274 A094689

Adjacent sequences:  A278386 A278387 A278388 * A278390 A278391 A278392

KEYWORD

nonn,cons,hard,more

AUTHOR

Jon E. Schoenfield, Nov 20 2016

EXTENSIONS

Edited and a(21)-a(26) from Robert G. Wilson v, Mar 07 2019

STATUS

approved

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Last modified May 12 23:35 EDT 2021. Contains 343829 sequences. (Running on oeis4.)