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#59 by N. J. A. Sloane at Sun Feb 25 11:10:05 EST 2024
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#58 by M. F. Hasler at Tue Feb 20 18:42:00 EST 2024
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#57 by M. F. Hasler at Tue Feb 20 18:35:18 EST 2024
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| COMMENTS
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The above mentioned average functions can also be written g(k) = f(k) + (-1)^k/prime(k)/2 and h(k) = g(k) + (-1)^k (1/prime(k) - 1/prime(k-1))/4 = f(k) + (-1)^k (3/prime(k) - 1/prime(k-1))/4. - M. F. Hasler, Feb 20 2024
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| FORMULA
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c = lim_{n -> oo} A024530(n)/A002110(n). - M. F. Hasler, Feb 20 2024
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| PROG
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(PARI) L=2^N=1; h=List([1/4, 1/6 + S=.5-1/o=3]); forprime(p=o+1, oo, S+=(-1)^L/p; L--|| print([L=2^N++, p, S, listput(h, S+(3/p-1/o)/4)]); o=p) \\ in PARI version > 2.13 listput() may not return the element so one must add +h[#h]
A(x, n=#x)=(x[n]*x[n-2]-x[n-1]^2)/(x[n]+x[n-2]-2*x[n-1]) \\ This is Aitken's Delta-square extrapolation for the last 3 elements of the list x. One can check that the extrapolation is useful for the sequence of raw partial sums (f(2^k)), but not for the smooth/average sequence (h(2^k)). - M. F. Hasler, Feb 20 2024
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| CROSSREFS
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Cf. A024530 (numerator of partial sums), A002110 (denominators: primorials).
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| STATUS
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approved
editing
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#56 by Alois P. Heinz at Fri Oct 14 15:28:19 EDT 2022
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| KEYWORD
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cons,hard,more,nonn,changed
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| STATUS
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editing
approved
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#55 by Alois P. Heinz at Fri Oct 14 15:27:51 EDT 2022
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#54 by Hugo Pfoertner at Fri Oct 14 15:19:52 EDT 2022
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Discussion
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| Fri Oct 14
| 15:27
| Alois P. Heinz: please do not do this again ... if you don't understand the system then do not change the system ...
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#53 by Hugo Pfoertner at Fri Oct 14 15:16:32 EDT 2022
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| LINKS
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Nicholas Juricic, <a href="/A078437/b078437.txt">Table of n, a(n) for n = 0..13</a>
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| STATUS
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proposed
editing
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#52 by Nicholas Juricic at Thu Oct 13 20:26:06 EDT 2022
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Discussion
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| Fri Oct 14
| 01:09
| Michel Marcus: are you sure ? I ask this because for instance A078416 does have what you say
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| 10:13
| Nicholas Juricic: When I click on "list" for A078437, I see it like I described: the left column of the table contains no positive terms. As you say, the same is true for A078416. Upon checking this morning, I see the same is true for A000796. (The b-bile for A000796 has n being positive, still.)
My initial thinking was along lines "it doesn't make sense to talk about the negative first, negative second, etc., digit of these numbers". Although, it could make sense to have n correspond to the power of 10 in the decimal expansion; in that case, the offset seems mistaken in the sequences we've mentioned. E.g. A000796 has in "list" n = 1 a(n) = 3, whereas pi equal to 3.14..., not 31.4...
I am most grateful for your thoughts.
(Also, I didn't mean for the comment line beginning with # to show up in the b-file, and was not sure if I should reupload it without the comment line, since we are rather discussing it here.) Thank you!
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| 15:19
| Hugo Pfoertner: This is a totally inappropriate place to rant about b-file definitions of constants. We have thousands of b-files for constants where this works fine. If you have something to discuss about this, please do so on the SeqFan mailing list.
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#51 by Nicholas Juricic at Thu Oct 13 20:25:14 EDT 2022
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| LINKS
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Nicholas Juricic, <a href="/A078437/b078437.txt">Table of n, a(n) for n = 0..13</a>
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#50 by Nicholas Juricic at Thu Oct 13 20:23:33 EDT 2022
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| LINKS
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Nicholas Juricic, <a href="/A078437/b078437.texttxt">Table of n, a(n) for n = MIN0..MAX13</a>
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