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Problems & Puzzles:
Puzzles
Problem 90.
Find the first 8 in...
On March 29, 2025 my friend G. L. Honaker, Jr. has
published the following Curio/Challenge-Reward and
problem:
G. L. Honaker, Jr. offers
$8 to the person or team that finds the first occurrence
of digit 8 in the constant
1/2 - 1/3 + 1/5 -
1/7 + 1/11 - 1/13 + ... = 0.269... .
(See the Curio
here).
"...(this) alternating
series
converges (Robinson and
Potter 1971)".
See A078437
and
this.
Q.
Please send your decimal expression of this alternating
sum, up to the first asked decimal (8), indicating the
largest prime needed to get it.
Important Note: Regarding this challenge, I will be just an
intermediate between Honaker and the participants. While I will
annotate the day and the hour I receive your email/answers, he
alone will decide who is the winner.
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On May-1-2025, 2:08 am (México, central time),
Simon Cavegn sent his solution to this Problem:
Continued h(k) from https://oeis.org/A078437
h(k) =
k (f(k-2) + 2*f(k-1) + f(k))/4
========== ============================
2 0.29166666666666666...
4 0.28095238095238095...
8 0.26875529011751921...
16 0.27058892362329746...
32 0.27009944617052797...
64 0.26963971020080367...
128 0.26959147218377685...
256 0.26959653902072193...
512 0.26960402179695026...
1024 0.26960568606633210...
2048 0.26960649673621509...
4096 0.26960645080540929...
8192 0.26960627432070023...
16384 0.26960633643086948...
32768 0.26960634835658329...
65536 0.26960635083481533...
131072 0.26960635144743392...
262144 0.26960635199009778...
524288 0.26960635199971603...
1048576 0.26960635195886861...
2097152 0.26960635197214933...
4194304 0.26960635197019215...
8388608 0.26960635197186919...
16777216 0.26960635197171149...
33554432 0.26960635197146884...
67108864 0.26960635197167534...
134217728 0.26960635197167145...
268435456 0.26960635197166927...
536870912 0.26960635197167200...
1073741824 0.26960635197167416...
2147483648 0.26960635197167454...
4294967296 0.26960635197167462...
8589934592
0.2696063519716746115951753723391177045452704199650018570665...
17179869184 0.2696063519716745911151644809251657437243751753378500233783...
34359738368 0.2696063519716745933571121943551630117783437944488045026659...
68719476736 0.2696063519716745948464031246903256452110028349813594133690...
137438953472 0.2696063519716745947477102160199100133802487290017570637917...
274877906944 0.2696063519716745949231122858949180938393568249345518460973...
549755813888 0.2696063519716745948577548318546254752594315752839685372454...
1099511627776 0.2696063519716745948490506428349040904990465420119594007519...
2199023255552 0.2696063519716745948520206964438658721432622448204777318986...
4398046511104 0.2696063519716745948516189025839897789324530848126518723457...
I calculated 3 different averages:
Average of last 3 values weighted by pascal triangle entry 1,2,1, this is
the h(k) function used by https://oeis.org/A078437
Average of last 8 values weighted by pascal triangle entry
1,7,21,35,35,21,7,1, named r(k)
Average of last 8 values weighted equally, named a(k)
h(k)=0.26960635197167459485161890258398977893245308481265187234570,
k=4398046511104
r(k)=0.26960635197167459485161890281395580110416961656337841996337,
k=4398046511104
a(k)=0.26960635197167459485161890353310749587938919254573665846864,
k=4398046511104
The constant's value : 0.2696063519716745948
Next digit is likely : 0.26960635197167459485
The calculation took 11 days and 10 hours on a Notebook CPU "AI 9 HX 370",
implemented in C# using https://gmplib.org/
***
and this is the comment by Mr. Honaker, Jr.:
Phi = 0.2696063519716745948... [Cavegn]
Simon,
Thank you very much for this result! I have submitted it to OEIS for
approval. If you email me your mailing address, I will be happy to send
you the promised reward money ($8US).
G. L. Honaker, Jr.
Content Editor: Prime
Curios!
***
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